4596:
7189:
4085:
4990:
6970:
4591:{\displaystyle {\begin{matrix}\operatorname {Proj} \left({\dfrac {\mathbb {C} }{(x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4})}}\right)&\to &\operatorname {Proj} \left({\dfrac {\mathbb {C} }{(x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+\varepsilon x_{0}^{a_{0}}x_{1}^{a_{1}}x_{2}^{a_{2}}x_{3}^{a_{3}})}}\right)\\\downarrow &&\downarrow \\\operatorname {Spec} (k)&\to &\operatorname {Spec} (k)\end{matrix}}}
5384:
1355:
4720:
7184:{\displaystyle {\begin{matrix}X&\to &{\mathfrak {X}}_{2}&\to &{\mathfrak {X}}_{3}&\to \cdots \\\downarrow &&\downarrow &&\downarrow &\\\operatorname {Spec} (\mathbb {F} _{p})&\to &\operatorname {Spec} (\mathbb {Z} /(p^{2}))&\to &\operatorname {Spec} (\mathbb {Z} /(p^{3}))&\to \cdots \end{matrix}}}
6959:
5241:
4074:
5230:
5929:
1159:
3017:
4985:{\displaystyle F(A)=\left\{{\begin{matrix}\operatorname {Proj} \left({\dfrac {\mathbb {C} }{(x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4})}}\right)&\to &{\mathfrak {X}}\\\downarrow &&\downarrow \\\operatorname {Spec} (k)&\to &\operatorname {Spec} (A)\end{matrix}}\right\}}
1392:
It should be clear there could be many deformations of a single germ of analytic functions. Because of this, there are some book-keeping devices required to organize all of this information. These organizational devices are constructed using tangent cohomology. This is formed by using the
2382:
6109:
6472:
6829:
2829:
around a point where lying above that point is the space of interest. It is typically the case that it is easier to describe the functor for a moduli problem instead of finding an actual space. For example, if we want to consider the moduli-space of hypersurfaces of degree
1064:
536:
1148:
5379:{\displaystyle {\begin{matrix}X&\to &{\mathfrak {X}}&\to &{\mathfrak {X}}'\\\downarrow &&\downarrow &&\downarrow \\\operatorname {Spec} (k)&\to &\operatorname {Spec} (A)&\to &\operatorname {Spec} (A')\end{matrix}}}
3797:
239:. These obviously depend on two parameters, a and b, whereas the isomorphism classes of such curves have only one parameter. Hence there must be an equation relating those a and b which describe isomorphic elliptic curves. It turns out that curves for which
1603:
1913:
6330:
3869:
65:
does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations. In some form these considerations have a history of centuries in mathematics, but also in
6818:
5139:
3592:
5756:
1350:{\displaystyle {\begin{matrix}{\frac {\mathbb {C} \{x,y\}}{(y^{2}-x^{n})}}&\leftarrow &{\frac {\mathbb {C} \{x,y,s\}}{(y^{2}-x^{n}+s)}}\\\uparrow &&\uparrow \\\mathbb {C} &\leftarrow &\mathbb {C} \{s\}\end{matrix}}}
5523:
2927:
2039:
837:
7248:
3283:
3180:
2233:
7481:
371:
These examples are the beginning of a theory applying to holomorphic families of complex manifolds, of any dimension. Further developments included: the extension by
Spencer of the techniques to other structures of
654:
2261:
6954:{\displaystyle {\begin{matrix}X&\to &{\mathfrak {X}}_{2}\\\downarrow &&\downarrow \\\operatorname {Spec} (\mathbb {F} _{p})&\to &\operatorname {Spec} (\mathbb {Z} /(p^{2}))\end{matrix}}}
5944:
6341:
7416:
424:, or germs of functions on a space. Grothendieck was the first to find this far-reaching generalization for deformations and developed the theory in that context. The general idea is there should exist a
5721:
2119:
2663:
7503:(roughly speaking, to formalise the idea that a string theory can be regarded as a deformation of a point-particle theory). This is now accepted as proved, after some hitches with early announcements.
982:
469:
4714:(which is topologically a point) allows us to organize this infinitesimal data. Since we want to consider all possible expansions, we will let our predeformation functor be defined on objects as
6346:
5949:
5761:
2791:
714:
6201:
6696:
2916:
1153:
where the horizontal arrows are isomorphisms. For example, there is a deformation of the plane curve singularity given by the opposite diagram of the commutative diagram of analytic algebras
3603:
1656:
2483:
4671:
1503:
1075:
4712:
4069:{\displaystyle {\begin{matrix}\operatorname {Proj} \left({\dfrac {\mathbb {C} }{(x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4})}}\right)\\\downarrow \\\operatorname {Spec} (k)\end{matrix}}}
1780:
6209:
5435:
455:
6655:
5634:
5128:
2547:
3067:
331:
7277:
6752:
2877:
2734:
1462:
195:
5225:{\displaystyle {\begin{matrix}X&\to &{\mathfrak {X}}\\\downarrow &&\downarrow \\\operatorname {Spec} (k)&\to &\operatorname {Spec} (A)\end{matrix}}}
3855:
In general, since we want to consider arbitrary order Taylor expansions in any number of variables, we will consider the category of all local artin algebras over a field.
5587:
5076:
3306:
3087:
3429:
2425:
6760:
6517:
5924:{\displaystyle {\begin{aligned}H^{1}(C,T_{C})&\cong H^{0}(C,T_{C}^{*}\otimes \omega _{C})^{\vee }\\&\cong H^{0}(C,\omega _{C}^{\otimes 2})^{\vee }\end{aligned}}}
1495:
6604:. So after several repetitions of the procedure, eventually we'll obtain a curve of genus 0, i.e. a rational curve. The existence and the properties of deformations of
3486:
3364:
3494:
1752:
1716:
907:
418:
6138:
6723:
3827:
2690:
974:
875:
6546:
2823:
3850:
2145:
581:
to complex manifolds and complex analytic spaces by considering the sheaves of germs of holomorphic functions, tangent spaces, etc. Such algebras are of the form
7747:
7338:
5744:
5415:
5033:
5013:
2848:
2253:
1936:
1772:
1680:
1415:
1378:
947:
927:
734:
3012:{\displaystyle F(S)=\left\{{\begin{matrix}X\\\downarrow \\S\end{matrix}}:{\text{ each fiber is a degree }}d{\text{ hypersurface in }}\mathbb {P} ^{n}\right\}}
4673:. Then, the space on the right hand corner is one example of an infinitesimal deformation: the extra scheme theoretic structure of the nilpotent elements in
5443:
1944:
742:
57:
Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of
7200:
3195:
3095:
3308:
term contains the derivative of the monomial, demonstrating its use in calculus. We could also interpret this equation as the first two terms of the
5532:. Since we are considering the tangent space of a point of some moduli space, we can define the tangent space of our (pre-)deformation functor as
2150:
7691:
7427:
7593:
2377:{\displaystyle T^{1}(A)\cong {\frac {A^{n}}{\left({\frac {\partial f}{\partial z_{1}}},\ldots ,{\frac {\partial f}{\partial z_{n}}}\right)}}}
587:
6620:
One of the major applications of deformation theory is in arithmetic. It can be used to answer the following question: if we have a variety
6576:
of positive dimension Mori showed that there is a rational curve passing through every point. The method of the proof later became known as
6104:{\displaystyle {\begin{aligned}h^{0}(C,\omega _{C}^{\otimes 2})-h^{1}(C,\omega _{C}^{\otimes 2})&=2(2g-2)-g+1\\&=3g-3\end{aligned}}}
553:, it is constructed as a quotient of the smooth curves in the Hilbert scheme. If the pullback square is not unique, then the family is only
6467:{\displaystyle {\begin{aligned}{\text{deg}}((\omega _{C}^{\otimes 2})^{\vee }\otimes \omega _{C})&=4-4g+2g-2\\&=2-2g\end{aligned}}}
3034:
Infinitesimals have long been in use by mathematicians for non-rigorous arguments in calculus. The idea is that if we consider polynomials
565:
One of the useful and readily computable areas of deformation theory comes from the deformation theory of germs of complex spaces, such as
123:
42:, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of
7551:
7367:
5642:
2050:
2552:
1059:{\displaystyle {\begin{matrix}X_{0}&\to &X\\\downarrow &&\downarrow \\*&{\xrightarrow{}}&S\end{matrix}}}
7739:
531:{\displaystyle {\begin{matrix}X&\to &{\mathfrak {X}}\\\downarrow &&\downarrow \\S&\to &B\end{matrix}}}
7668:
6597:
380:, with a consequent substantive clarification of earlier work; and deformation theory of other structures, such as algebras.
7823:
7660:
7641:
1360:
In fact, Milnor studied such deformations, where a singularity is deformed by a constant, hence the fiber over a non-zero
7358:
Another application of deformation theory is with Galois deformations. It allows us to answer the question: If we have a
2753:
662:
212:
of the same sheaf; which is always zero in case of a curve, for general reasons of dimension. In the case of genus 0 the
6143:
3438:
Moreover, if we want to consider higher-order terms of a Taylor approximation then we could consider the artin algebras
7541:
3792:{\displaystyle f(x)=f(0)+{\frac {f^{(1)}(0)}{1!}}x+{\frac {f^{(2)}(0)}{2!}}x^{2}+{\frac {f^{(3)}(0)}{3!}}x^{3}+\cdots }
7818:
7681:
7636:
6660:
246:
has the same value, describe isomorphic curves. I.e. varying a and b is one way to deform the structure of the curve
2885:
1394:
7778:
1611:
7546:
5603:
87:
1598:{\displaystyle \cdots \xrightarrow {s} R_{-2}\xrightarrow {s} R_{-1}\xrightarrow {s} R_{0}\xrightarrow {p} A\to 0}
1143:{\displaystyle {\begin{matrix}X'&\to &X\\\downarrow &&\downarrow \\S'&\to &S\end{matrix}}}
7526:
5935:
2433:
365:
4604:
4079:
If we want to consider an infinitesimal deformation of this space, then we could write down a
Cartesian square
1908:{\displaystyle 0\to T^{0}(A)\to {\text{Der}}(R_{0})\xrightarrow {d} {\text{Hom}}_{R_{0}}(I,A)\to T^{1}(A)\to 0}
574:
47:
7516:
6325:{\displaystyle h^{1}(C,\omega _{C}^{\otimes 2})=h^{0}(C,(\omega _{C}^{\otimes 2})^{\vee }\otimes \omega _{C})}
4676:
6609:
7288:
3863:
To motivate the definition of a pre-deformation functor, consider the projective hypersurface over a field
5420:
2705:
430:
353:
95:
2488:
hence the only deformations are given by adding constants or linear factors, so a general deformation of
736:
is an ideal. For example, many authors study the germs of functions of a singularity, such as the algebra
7740:"Perturbations, Deformations, and Variations (and "Near-Misses" in Geometry, Physics, and Number Theory"
7496:
7359:
6623:
6585:
5608:
2491:
373:
127:
51:
43:
3037:
287:
7797:
7253:
6728:
2853:
2710:
1428:
163:
3432:
421:
357:
7648:
6813:{\displaystyle {\begin{matrix}X\\\downarrow \\\operatorname {Spec} (\mathbb {F} _{p})\end{matrix}}}
6557:
5084:
356:
on a
Riemann surface, again something known classically. The dimension of the moduli space, called
111:
91:
75:
5538:
3291:
3072:
7631:
7492:
3587:{\displaystyle (x+\varepsilon )^{3}=x^{3}+3x^{2}\varepsilon +3x\varepsilon ^{2}+\varepsilon ^{3}}
3369:
2390:
201:
6480:
1467:
3441:
3319:
153:
theory. The general
Kodaira–Spencer theory identifies as the key to the deformation theory the
7664:
7589:
7536:
7299:
6569:
3313:
1721:
1685:
880:
391:
150:
122:, after deformation techniques had received a great deal of more tentative application in the
119:
6117:
5050:
7756:
7581:
7504:
7353:
6593:
3309:
570:
550:
337:
154:
115:
107:
7770:
6701:
3805:
2668:
952:
853:
7766:
7609:
6522:
2799:
461:
138:
376:; the assimilation of the Kodaira–Spencer theory into the abstract algebraic geometry of
126:. One expects, intuitively, that deformation theory of the first order should equate the
3832:
2127:
1397:, and potentially modifying it by adding additional generators for non-regular algebras
7307:
7292:
6565:
6561:
5729:
5400:
5018:
4998:
2833:
2238:
1921:
1757:
1665:
1422:
1400:
1363:
932:
912:
719:
566:
542:
349:
223:
which is therefore 1. It is known that all curves of genus one have equations of form
205:
146:
142:
7812:
7500:
7195:
5747:
5518:{\displaystyle TX:=\operatorname {Hom} _{{\text{Sch}}/k}(\operatorname {Spec} (k),X)}
2740:
274:
28:
6519:
for line bundles of negative degree. Therefore the dimension of the moduli space is
5636:
can be deduced using elementary deformation theory. Its dimension can be computed as
2034:{\displaystyle {\frac {\mathbb {C} \{z_{1},\ldots ,z_{n}\}}{(f_{1},\ldots ,f_{m})}}}
7717:
7652:
6573:
2826:
832:{\displaystyle A\cong {\frac {\mathbb {C} \{z_{1},\ldots ,z_{n}\}}{(y^{2}-x^{n})}}}
377:
217:
131:
7761:
7696:
5389:
the name smooth comes from the lifting criterion of a smooth morphism of schemes.
5235:
does there exist an extension of this cartesian diagram to the cartesian diagrams
3488:. For our monomial, suppose we want to write out the second order expansion, then
2825:
is a point. The idea is that we want to study the infinitesimal structure of some
1497:
is a surjective map of analytic algebras, and this map fits into an exact sequence
7243:{\displaystyle {\mathfrak {X}}=\operatorname {Spet} ({\mathfrak {X}}_{\bullet })}
5078:
such that the square of any element in the kernel is zero, there is a surjection
3278:{\displaystyle (x+\varepsilon )^{3}=x^{3}+3x^{2}\varepsilon +O(\varepsilon ^{2})}
3175:{\displaystyle F(x,\varepsilon )\equiv f(x)+\varepsilon g(x)+O(\varepsilon ^{2})}
7735:
7585:
7521:
5529:
3366:
we see that arguments with infinitesimals can work. This motivates the notation
546:
71:
20:
7657:
Deformation Theory and
Quantum Groups with Applications to Mathematical Physics
5746:
because the deformation space is the tangent space of the moduli space. Using
2737:
3022:
Although in general, it is more convenient/required to work with functors of
2228:{\displaystyle f=(f_{1},\ldots ,f_{m}):\mathbb {C} ^{n}\to \mathbb {C} ^{m}}
1069:
These deformations have an equivalence relation given by commutative squares
134:. The phenomena turn out to be rather subtle, though, in the general case.
7580:. Encyclopaedia of Mathematical Sciences. Vol. 10. pp. 105–194.
6584:
through a chosen point and keep deforming it until it breaks into several
3089:, then only the first order terms really matter; that is, we can consider
7722:
7531:
3186:
3023:
7476:{\displaystyle G\to \operatorname {GL} _{n}(\mathbb {Z} _{p}){\text{?}}}
549:, or a quotient of one of them. For example, in the construction of the
2701:
649:{\displaystyle A\cong {\frac {\mathbb {C} \{z_{1},\ldots ,z_{n}\}}{I}}}
67:
1044:
1037:
846:
is then an object in the opposite category of such algebras. Then, a
7507:
is among those who have offered a generally accepted proof of this.
3802:
hence the previous two equations show that the second derivative of
3185:
A simple application of this is that we can find the derivatives of
1837:
1580:
1560:
1537:
1514:
1417:. In the case of analytic algebras these resolutions are called the
106:
The most salient deformation theory in mathematics has been that of
6592:
by one of the components has the effect of decreasing either the
5133:
This is motivated by the following question: given a deformation
7499:) stimulated much interest in deformation theory in relation to
3597:
Recall that a Taylor expansion (at zero) can be written out as
7411:{\displaystyle G\to \operatorname {GL} _{n}(\mathbb {F} _{p})}
1608:
Then, by taking the differential graded module of derivations
149:
has a one-parameter family of complex structures, as shown in
6608:
require arguments from deformation theory and a reduction to
352:). In other words, deformations are regulated by holomorphic
5716:{\displaystyle \dim({\mathcal {M}}_{g})=\dim H^{1}(C,T_{C})}
5658:
5615:
2114:{\displaystyle T^{1}(A)\cong {\frac {A^{m}}{df\cdot A^{n}}}}
3312:
of the monomial. Infinitesimals can be made rigorous using
2700:
Another method for formalizing deformation theory is using
2658:{\displaystyle F(x,y,a_{1},a_{2})=y^{2}-x^{3}+a_{1}+a_{2}x}
2235:. For example, the deformations of a hypersurface given by
1425:. This is a graded-commutative differential graded algebra
7723:
MSRI – Deformation Theory and Moduli in
Algebraic Geometry
82:
was recognised, with the topological interpretation of an
114:. This was put on a firm basis by foundational work of
6975:
6964:
then we can always extend it to a diagram of the form
6834:
6765:
6725:
implies that every deformation induces a variety over
5246:
5144:
4744:
4090:
3874:
2952:
1754:
contains information about all of the deformations of
1421:
for the mathematician who first studied such objects,
1164:
1080:
987:
474:
50:. The name is an analogy to non-rigid structures that
7430:
7370:
7310:
7256:
7203:
6973:
6832:
6763:
6731:
6704:
6663:
6626:
6525:
6483:
6344:
6212:
6146:
6120:
5947:
5759:
5732:
5645:
5611:
5541:
5446:
5423:
5403:
5244:
5142:
5087:
5053:
5021:
5001:
4758:
4723:
4679:
4607:
4270:
4104:
4088:
3888:
3872:
3835:
3808:
3606:
3497:
3444:
3372:
3322:
3294:
3198:
3098:
3075:
3040:
2930:
2888:
2856:
2836:
2802:
2756:
2713:
2671:
2555:
2494:
2436:
2393:
2264:
2241:
2153:
2130:
2053:
1947:
1924:
1783:
1760:
1724:
1688:
1668:
1614:
1506:
1470:
1431:
1403:
1366:
1162:
1078:
985:
955:
935:
915:
883:
877:
is given by a flat map of germs of analytic algebras
856:
745:
722:
665:
590:
472:
433:
394:
388:
The most general form of a deformation is a flat map
290:
166:
7718:
Notes from
Hartshorne's Course on Deformation Theory
7576:
Palamodov (1990). "Deformations of
Complex Spaces".
7291:
asserts, roughly speaking, that the deformations of
3026:
instead of sets. This is true for moduli of curves.
1774:
and can be readily computed using the exact sequence
266:
actually change the isomorphism class of the curve.
141:, one can explain that the complex structure on the
2786:{\displaystyle F:{\text{Art}}_{k}\to {\text{Sets}}}
709:{\displaystyle \mathbb {C} \{z_{1},\ldots ,z_{n}\}}
7475:
7410:
7332:
7271:
7242:
7183:
6953:
6812:
6746:
6717:
6690:
6649:
6540:
6511:
6466:
6324:
6196:{\displaystyle h^{1}(C,\omega _{C}^{\otimes 2})=0}
6195:
6132:
6103:
5923:
5738:
5715:
5628:
5581:
5517:
5429:
5409:
5378:
5224:
5122:
5070:
5027:
5007:
4984:
4706:
4665:
4590:
4068:
3844:
3821:
3791:
3586:
3480:
3423:
3358:
3300:
3277:
3174:
3081:
3061:
3011:
2910:
2871:
2842:
2817:
2785:
2728:
2684:
2657:
2541:
2477:
2419:
2376:
2247:
2227:
2139:
2113:
2033:
1930:
1907:
1766:
1746:
1710:
1674:
1650:
1597:
1489:
1456:
1409:
1372:
1349:
1142:
1058:
968:
941:
921:
901:
869:
831:
728:
708:
648:
530:
449:
412:
325:
189:
61:, in that varying a solution may not be possible,
541:In many cases, this universal family is either a
216:vanishes, also. For genus 1 the dimension is the
6698:? If our variety is a curve, then the vanishing
6691:{\displaystyle {\mathfrak {X}}/\mathbb {Z} _{p}}
2911:{\displaystyle F:{\text{Sch}}\to {\text{Sets}}}
1651:{\displaystyle ({\text{Der}}(R_{\bullet }),d)}
31:conditions associated with varying a solution
7748:Bulletin of the American Mathematical Society
6580:. The rough idea is to start with some curve
35:of a problem to slightly different solutions
8:
1988:
1956:
1388:Cohomological Interpretation of deformations
1340:
1334:
1257:
1239:
1187:
1175:
792:
760:
703:
671:
637:
605:
457:such that any deformation can be found as a
6556:Deformation theory was famously applied in
716:is the ring of convergent power-series and
7692:Course Notes on Deformation Theory (Artin)
5397:Recall that the tangent space of a scheme
2478:{\displaystyle {\frac {A^{2}}{(y,x^{2})}}}
842:representing a plane-curve singularity. A
561:Deformations of germs of analytic algebras
269:One can go further with the case of genus
94:also looks at deformations, in general of
7760:
7468:
7459:
7455:
7454:
7441:
7429:
7421:how can we extend it to a representation
7399:
7395:
7394:
7381:
7369:
7321:
7309:
7263:
7259:
7258:
7255:
7231:
7225:
7224:
7205:
7204:
7202:
7157:
7145:
7141:
7140:
7112:
7100:
7096:
7095:
7070:
7066:
7065:
7017:
7011:
7010:
6996:
6990:
6989:
6974:
6972:
6935:
6923:
6919:
6918:
6893:
6889:
6888:
6855:
6849:
6848:
6833:
6831:
6797:
6793:
6792:
6764:
6762:
6738:
6734:
6733:
6730:
6709:
6703:
6682:
6678:
6677:
6671:
6665:
6664:
6662:
6641:
6637:
6636:
6630:
6625:
6524:
6488:
6482:
6395:
6382:
6369:
6364:
6349:
6345:
6343:
6313:
6300:
6287:
6282:
6260:
6241:
6236:
6217:
6211:
6175:
6170:
6151:
6145:
6119:
6023:
6018:
5999:
5980:
5975:
5956:
5948:
5946:
5911:
5898:
5893:
5874:
5854:
5844:
5831:
5826:
5807:
5787:
5768:
5760:
5758:
5731:
5704:
5685:
5663:
5657:
5656:
5644:
5620:
5614:
5613:
5610:
5546:
5540:
5466:
5461:
5460:
5445:
5422:
5402:
5275:
5274:
5260:
5259:
5245:
5243:
5158:
5157:
5143:
5141:
5086:
5052:
5020:
5000:
4914:
4913:
4889:
4884:
4871:
4866:
4853:
4848:
4835:
4830:
4812:
4799:
4786:
4773:
4762:
4761:
4757:
4743:
4722:
4678:
4666:{\displaystyle a_{0}+a_{1}+a_{2}+a_{3}=4}
4651:
4638:
4625:
4612:
4606:
4502:
4497:
4492:
4480:
4475:
4470:
4458:
4453:
4448:
4436:
4431:
4426:
4410:
4405:
4392:
4387:
4374:
4369:
4356:
4351:
4324:
4311:
4298:
4285:
4274:
4273:
4269:
4235:
4230:
4217:
4212:
4199:
4194:
4181:
4176:
4158:
4145:
4132:
4119:
4108:
4107:
4103:
4089:
4087:
4019:
4014:
4001:
3996:
3983:
3978:
3965:
3960:
3942:
3929:
3916:
3903:
3892:
3891:
3887:
3873:
3871:
3834:
3813:
3807:
3777:
3741:
3734:
3725:
3689:
3682:
3644:
3637:
3605:
3578:
3565:
3543:
3527:
3514:
3496:
3469:
3457:
3443:
3412:
3400:
3371:
3347:
3335:
3321:
3293:
3266:
3244:
3228:
3215:
3197:
3163:
3097:
3074:
3039:
2998:
2994:
2993:
2987:
2979:
2951:
2929:
2903:
2895:
2887:
2863:
2859:
2858:
2855:
2835:
2801:
2778:
2769:
2764:
2755:
2720:
2715:
2712:
2676:
2670:
2646:
2633:
2620:
2607:
2591:
2578:
2554:
2533:
2520:
2493:
2463:
2443:
2437:
2435:
2411:
2398:
2392:
2358:
2340:
2322:
2304:
2293:
2287:
2269:
2263:
2240:
2219:
2215:
2214:
2204:
2200:
2199:
2186:
2167:
2152:
2129:
2102:
2082:
2076:
2058:
2052:
2019:
2000:
1982:
1963:
1952:
1951:
1948:
1946:
1923:
1884:
1854:
1849:
1844:
1824:
1812:
1794:
1782:
1759:
1729:
1723:
1693:
1687:
1667:
1630:
1618:
1613:
1570:
1547:
1524:
1505:
1475:
1469:
1439:
1430:
1402:
1365:
1330:
1329:
1318:
1317:
1282:
1269:
1235:
1234:
1231:
1212:
1199:
1171:
1170:
1167:
1163:
1161:
1079:
1077:
1038:
1032:
994:
986:
984:
960:
954:
934:
914:
882:
861:
855:
817:
804:
786:
767:
756:
755:
752:
744:
721:
697:
678:
667:
666:
664:
631:
612:
601:
600:
597:
589:
488:
487:
473:
471:
435:
434:
432:
393:
308:
295:
289:
186:
171:
165:
145:is isolated (no moduli). For genus 1, an
54:slightly to accommodate external forces.
7495:arising in the context of algebras (and
4707:{\displaystyle \operatorname {Spec} (k)}
7563:
5726:for an arbitrary smooth curve of genus
7697:Studying Deformation Theory of Schemes
3030:Technical remarks about infinitesimals
1682:. These cohomology groups are denoted
7614:Higher-Dimensional Algebraic Geometry
7298:is controlled by deformations of the
7194:This implies that we can construct a
6754:; that is, if we have a smooth curve
3316:in local artin algebras. In the ring
2879:, then we could consider the functor
7:
7612:(2001). "3. Bend-and-Break Lemmas".
7571:
7569:
7567:
5430:{\displaystyle \operatorname {Hom} }
5043:A pre-deformation functor is called
450:{\displaystyle {\mathfrak {X}}\to B}
124:Italian school of algebraic geometry
7226:
7206:
7012:
6991:
6850:
6666:
6657:, what are the possible extensions
5602:One of the first properties of the
5276:
5261:
5159:
4915:
489:
436:
7780:Why deformations are cohomological
7322:
6650:{\displaystyle X/\mathbb {F} _{p}}
5750:the tangent space is isomorphic to
5629:{\displaystyle {\mathcal {M}}_{g}}
5593:Applications of deformation theory
2981: each fiber is a degree
2542:{\displaystyle f(x,y)=y^{2}-x^{3}}
2351:
2343:
2315:
2307:
2044:then its deformations are equal to
305:
180:
14:
7805:, lecture notes by Brian Osserman
7798:"A glimpse of deformation theory"
7704:Deformations of Algebraic Schemes
7552:Degeneration (algebraic geometry)
3062:{\displaystyle F(x,\varepsilon )}
1662:of the germ of analytic algebras
326:{\displaystyle H^{0}(\Omega ^{})}
208:. There is an obstruction in the
102:Deformations of complex manifolds
7272:{\displaystyle \mathbb {Z} _{p}}
6747:{\displaystyle \mathbb {Z} _{p}}
5528:where the source is the ring of
2872:{\displaystyle \mathbb {P} ^{n}}
2729:{\displaystyle {\text{Art}}_{k}}
1457:{\displaystyle (R_{\bullet },s)}
204:of sections of) the holomorphic
190:{\displaystyle H^{1}(\Theta )\,}
7283:Deformations of abelian schemes
5039:Smooth pre-deformation functors
850:of a germ of analytic algebras
577:. Note that this theory can be
7682:Deformations of complex spaces
7465:
7450:
7434:
7405:
7390:
7374:
7327:
7314:
7237:
7220:
7171:
7166:
7163:
7150:
7137:
7126:
7121:
7118:
7105:
7092:
7081:
7076:
7061:
7047:
7041:
7035:
7025:
7004:
6983:
6944:
6941:
6928:
6915:
6904:
6899:
6884:
6871:
6865:
6842:
6803:
6788:
6775:
6500:
6494:
6401:
6379:
6357:
6354:
6319:
6297:
6275:
6266:
6250:
6223:
6184:
6157:
6060:
6045:
6032:
6005:
5989:
5962:
5908:
5880:
5851:
5813:
5793:
5774:
5710:
5691:
5669:
5652:
5573:
5570:
5564:
5558:
5512:
5503:
5500:
5494:
5488:
5479:
5369:
5358:
5347:
5342:
5336:
5325:
5320:
5314:
5301:
5295:
5289:
5268:
5254:
5215:
5209:
5198:
5193:
5187:
5174:
5168:
5152:
5117:
5111:
5105:
5102:
5091:
5062:
4971:
4965:
4954:
4949:
4943:
4930:
4924:
4908:
4895:
4823:
4818:
4766:
4733:
4727:
4701:
4698:
4692:
4686:
4581:
4578:
4572:
4566:
4555:
4550:
4544:
4531:
4525:
4510:
4344:
4339:
4333:
4330:
4278:
4254:
4241:
4169:
4164:
4112:
4059:
4053:
4040:
4025:
3953:
3948:
3896:
3759:
3753:
3748:
3742:
3707:
3701:
3696:
3690:
3662:
3656:
3651:
3645:
3631:
3625:
3616:
3610:
3511:
3498:
3475:
3462:
3454:
3448:
3418:
3405:
3397:
3391:
3382:
3376:
3353:
3340:
3332:
3326:
3272:
3259:
3212:
3199:
3169:
3156:
3147:
3141:
3129:
3123:
3114:
3102:
3056:
3044:
2962:
2940:
2934:
2900:
2812:
2806:
2775:
2597:
2559:
2510:
2498:
2469:
2450:
2281:
2275:
2210:
2192:
2160:
2070:
2064:
2025:
1993:
1899:
1896:
1890:
1877:
1874:
1862:
1830:
1817:
1809:
1806:
1800:
1787:
1741:
1735:
1705:
1699:
1645:
1636:
1623:
1615:
1589:
1481:
1451:
1432:
1324:
1310:
1304:
1294:
1262:
1226:
1218:
1192:
1128:
1111:
1105:
1093:
1020:
1014:
1002:
893:
823:
797:
516:
504:
498:
482:
441:
404:
360:in this case, is computed as 3
320:
315:
309:
301:
183:
177:
1:
7762:10.1090/S0273-0979-04-01024-9
7661:American Mathematical Society
7487:Relationship to string theory
5598:Dimension of moduli of curves
5123:{\displaystyle F(A')\to F(A)}
976:fits into the pullback square
340:and the notation Ω means the
7578:Several Complex Variables IV
5582:{\displaystyle T_{F}:=F(k).}
3301:{\displaystyle \varepsilon }
3082:{\displaystyle \varepsilon }
2692:are deformation parameters.
1938:is isomorphic to the algebra
420:of complex-analytic spaces,
262:, but not all variations of
7637:Encyclopedia of Mathematics
7586:10.1007/978-3-642-61263-3_3
3424:{\displaystyle k=k/(y^{2})}
2989: hypersurface in
2420:{\displaystyle y^{2}-x^{3}}
1658:, its cohomology forms the
336:where Ω is the holomorphic
90:) around a given solution.
7840:
7688:(very down to earth intro)
7547:Moduli of algebraic curves
7351:
6564:to study the existence of
6512:{\displaystyle h^{0}(L)=0}
5604:moduli of algebraic curves
2147:is the jacobian matrix of
1490:{\displaystyle R_{0}\to A}
929:has a distinguished point
575:complex analytic varieties
384:Deformations and flat maps
78:a class of results called
46:to solving a problem with
7616:. Universitext. Springer.
3481:{\displaystyle k/(y^{k})}
3359:{\displaystyle k/(y^{2})}
844:germ of analytic algebras
200:where Θ is (the sheaf of
5417:can be described as the
2747:is defined as a functor
1747:{\displaystyle T^{1}(A)}
1711:{\displaystyle T^{k}(A)}
902:{\displaystyle f:X\to S}
413:{\displaystyle f:X\to S}
7680:Palamodov, V. P., III.
7542:Gromov–Witten invariant
7344:-power torsion points.
6616:Arithmetic deformations
6610:positive characteristic
6133:{\displaystyle g\geq 2}
5071:{\displaystyle A'\to A}
2745:pre-deformation functor
354:quadratic differentials
7527:Schlessinger's theorem
7477:
7412:
7334:
7273:
7244:
7185:
6955:
6814:
6748:
6719:
6692:
6651:
6542:
6513:
6475:
6468:
6333:
6326:
6197:
6134:
6112:
6105:
5932:
5925:
5740:
5724:
5717:
5630:
5583:
5519:
5431:
5411:
5380:
5226:
5124:
5072:
5047:if for any surjection
5029:
5009:
4986:
4708:
4667:
4592:
4070:
3846:
3823:
3793:
3588:
3482:
3431:, which is called the
3425:
3360:
3302:
3279:
3189:using infinitesimals:
3176:
3083:
3069:with an infinitesimal
3063:
3013:
2912:
2873:
2844:
2819:
2787:
2730:
2696:Functorial description
2686:
2659:
2543:
2486:
2479:
2421:
2385:
2378:
2249:
2229:
2141:
2122:
2115:
2042:
2035:
1932:
1916:
1909:
1768:
1748:
1712:
1676:
1652:
1606:
1599:
1491:
1458:
1411:
1395:Koszul–Tate resolution
1374:
1358:
1351:
1151:
1144:
1067:
1060:
1045:
970:
943:
923:
903:
871:
840:
833:
730:
710:
657:
650:
539:
532:
451:
414:
327:
191:
74:. For example, in the
7497:Hochschild cohomology
7478:
7413:
7360:Galois representation
7352:Further information:
7335:
7274:
7245:
7186:
6956:
6815:
6749:
6720:
6718:{\displaystyle H^{2}}
6693:
6652:
6578:Mori's bend-and-break
6543:
6514:
6469:
6337:
6327:
6205:
6198:
6135:
6106:
5940:
5926:
5752:
5741:
5718:
5638:
5631:
5584:
5520:
5432:
5412:
5381:
5227:
5125:
5073:
5030:
5010:
4987:
4709:
4668:
4593:
4071:
3847:
3824:
3822:{\displaystyle x^{3}}
3794:
3589:
3483:
3426:
3361:
3303:
3280:
3177:
3084:
3064:
3014:
2913:
2874:
2845:
2820:
2788:
2731:
2687:
2685:{\displaystyle a_{i}}
2660:
2544:
2480:
2429:
2422:
2379:
2257:
2250:
2230:
2142:
2116:
2046:
2036:
1940:
1933:
1910:
1776:
1769:
1749:
1713:
1677:
1653:
1600:
1499:
1492:
1459:
1412:
1375:
1352:
1155:
1145:
1071:
1061:
1033:
978:
971:
969:{\displaystyle X_{0}}
944:
924:
904:
872:
870:{\displaystyle X_{0}}
834:
738:
731:
711:
651:
583:
533:
465:
452:
415:
374:differential geometry
328:
192:
128:Zariski tangent space
44:differential calculus
16:Branch of mathematics
7824:Differential algebra
7686:Complex Variables IV
7649:Gerstenhaber, Murray
7428:
7368:
7308:
7254:
7250:giving a curve over
7201:
6971:
6830:
6761:
6729:
6702:
6661:
6624:
6541:{\displaystyle 3g-3}
6523:
6481:
6342:
6210:
6144:
6118:
6114:For curves of genus
5945:
5936:Riemann–Roch theorem
5757:
5730:
5643:
5609:
5539:
5444:
5421:
5401:
5242:
5140:
5085:
5051:
5019:
4999:
4721:
4677:
4605:
4086:
3870:
3833:
3806:
3604:
3495:
3442:
3433:ring of dual numbers
3370:
3320:
3292:
3196:
3096:
3073:
3038:
2928:
2886:
2854:
2834:
2818:{\displaystyle F(k)}
2800:
2754:
2711:
2669:
2553:
2492:
2434:
2391:
2387:For the singularity
2262:
2255:has the deformations
2239:
2151:
2128:
2051:
1945:
1922:
1781:
1758:
1722:
1686:
1666:
1612:
1504:
1468:
1429:
1401:
1364:
1160:
1076:
983:
953:
933:
913:
881:
854:
743:
720:
663:
588:
470:
431:
392:
366:Riemann–Roch theorem
288:
164:
7710:Hartshorne, Robin,
7517:Kodaira–Spencer map
7348:Galois deformations
6558:birational geometry
6377:
6295:
6249:
6183:
6031:
5988:
5906:
5836:
4894:
4876:
4858:
4840:
4509:
4487:
4465:
4443:
4415:
4397:
4379:
4361:
4240:
4222:
4204:
4186:
4024:
4006:
3988:
3970:
1841:
1584:
1564:
1541:
1518:
1043:
112:algebraic varieties
92:Perturbation theory
76:geometry of numbers
7819:Algebraic geometry
7712:Deformation Theory
7702:Sernesi, Eduardo,
7493:Deligne conjecture
7473:
7408:
7340:consisting of its
7330:
7289:Serre–Tate theorem
7269:
7240:
7181:
7179:
6951:
6949:
6823:and a deformation
6810:
6808:
6744:
6715:
6688:
6647:
6538:
6509:
6464:
6462:
6360:
6322:
6278:
6232:
6193:
6166:
6130:
6101:
6099:
6014:
5971:
5921:
5919:
5889:
5822:
5736:
5713:
5626:
5579:
5515:
5427:
5407:
5376:
5374:
5222:
5220:
5120:
5068:
5025:
5005:
4982:
4976:
4900:
4880:
4862:
4844:
4826:
4704:
4663:
4588:
4586:
4515:
4488:
4466:
4444:
4422:
4401:
4383:
4365:
4347:
4246:
4226:
4208:
4190:
4172:
4066:
4064:
4030:
4010:
3992:
3974:
3956:
3845:{\displaystyle 6x}
3842:
3819:
3789:
3584:
3478:
3421:
3356:
3314:nilpotent elements
3298:
3275:
3172:
3079:
3059:
3009:
2974:
2908:
2869:
2840:
2815:
2783:
2726:
2682:
2655:
2539:
2475:
2427:this is the module
2417:
2374:
2245:
2225:
2140:{\displaystyle df}
2137:
2111:
2031:
1928:
1905:
1764:
1744:
1708:
1672:
1660:tangent cohomology
1648:
1595:
1487:
1454:
1419:Tjurina resolution
1407:
1370:
1347:
1345:
1140:
1138:
1056:
1054:
966:
939:
919:
899:
867:
829:
726:
706:
646:
528:
526:
447:
410:
364:− 3, by the
323:
187:
80:isolation theorems
59:isolated solutions
25:deformation theory
7595:978-3-642-64766-6
7537:Cotangent complex
7471:
7333:{\displaystyle A}
6352:
5739:{\displaystyle g}
5464:
5410:{\displaystyle X}
5028:{\displaystyle k}
5015:is a local Artin
5008:{\displaystyle A}
4899:
4514:
4245:
4029:
3771:
3719:
3674:
2990:
2982:
2906:
2898:
2843:{\displaystyle d}
2781:
2767:
2718:
2473:
2372:
2365:
2329:
2248:{\displaystyle f}
2109:
2029:
1931:{\displaystyle A}
1847:
1842:
1815:
1767:{\displaystyle A}
1675:{\displaystyle A}
1621:
1585:
1565:
1542:
1519:
1410:{\displaystyle A}
1373:{\displaystyle s}
1298:
1222:
942:{\displaystyle 0}
922:{\displaystyle S}
827:
729:{\displaystyle I}
644:
571:complex manifolds
358:Teichmüller space
151:elliptic function
120:Donald C. Spencer
108:complex manifolds
7831:
7804:
7802:
7786:
7785:
7773:
7764:
7744:
7714:
7706:
7653:Stasheff, James
7645:
7618:
7617:
7610:Debarre, Olivier
7606:
7600:
7599:
7573:
7505:Maxim Kontsevich
7482:
7480:
7479:
7474:
7472:
7469:
7464:
7463:
7458:
7446:
7445:
7417:
7415:
7414:
7409:
7404:
7403:
7398:
7386:
7385:
7354:Deformation ring
7339:
7337:
7336:
7331:
7326:
7325:
7303:-divisible group
7278:
7276:
7275:
7270:
7268:
7267:
7262:
7249:
7247:
7246:
7241:
7236:
7235:
7230:
7229:
7210:
7209:
7190:
7188:
7187:
7182:
7180:
7162:
7161:
7149:
7144:
7117:
7116:
7104:
7099:
7075:
7074:
7069:
7051:
7045:
7039:
7022:
7021:
7016:
7015:
7001:
7000:
6995:
6994:
6960:
6958:
6957:
6952:
6950:
6940:
6939:
6927:
6922:
6898:
6897:
6892:
6869:
6860:
6859:
6854:
6853:
6819:
6817:
6816:
6811:
6809:
6802:
6801:
6796:
6753:
6751:
6750:
6745:
6743:
6742:
6737:
6724:
6722:
6721:
6716:
6714:
6713:
6697:
6695:
6694:
6689:
6687:
6686:
6681:
6675:
6670:
6669:
6656:
6654:
6653:
6648:
6646:
6645:
6640:
6634:
6547:
6545:
6544:
6539:
6518:
6516:
6515:
6510:
6493:
6492:
6473:
6471:
6470:
6465:
6463:
6441:
6400:
6399:
6387:
6386:
6376:
6368:
6353:
6350:
6331:
6329:
6328:
6323:
6318:
6317:
6305:
6304:
6294:
6286:
6265:
6264:
6248:
6240:
6222:
6221:
6202:
6200:
6199:
6194:
6182:
6174:
6156:
6155:
6139:
6137:
6136:
6131:
6110:
6108:
6107:
6102:
6100:
6078:
6030:
6022:
6004:
6003:
5987:
5979:
5961:
5960:
5930:
5928:
5927:
5922:
5920:
5916:
5915:
5905:
5897:
5879:
5878:
5863:
5859:
5858:
5849:
5848:
5835:
5830:
5812:
5811:
5792:
5791:
5773:
5772:
5745:
5743:
5742:
5737:
5722:
5720:
5719:
5714:
5709:
5708:
5690:
5689:
5668:
5667:
5662:
5661:
5635:
5633:
5632:
5627:
5625:
5624:
5619:
5618:
5588:
5586:
5585:
5580:
5551:
5550:
5524:
5522:
5521:
5516:
5475:
5474:
5470:
5465:
5462:
5436:
5434:
5433:
5428:
5416:
5414:
5413:
5408:
5385:
5383:
5382:
5377:
5375:
5368:
5299:
5293:
5284:
5280:
5279:
5265:
5264:
5231:
5229:
5228:
5223:
5221:
5172:
5163:
5162:
5129:
5127:
5126:
5121:
5101:
5077:
5075:
5074:
5069:
5061:
5034:
5032:
5031:
5026:
5014:
5012:
5011:
5006:
4991:
4989:
4988:
4983:
4981:
4977:
4928:
4919:
4918:
4905:
4901:
4898:
4893:
4888:
4875:
4870:
4857:
4852:
4839:
4834:
4821:
4817:
4816:
4804:
4803:
4791:
4790:
4778:
4777:
4765:
4759:
4713:
4711:
4710:
4705:
4672:
4670:
4669:
4664:
4656:
4655:
4643:
4642:
4630:
4629:
4617:
4616:
4597:
4595:
4594:
4589:
4587:
4529:
4520:
4516:
4513:
4508:
4507:
4506:
4496:
4486:
4485:
4484:
4474:
4464:
4463:
4462:
4452:
4442:
4441:
4440:
4430:
4414:
4409:
4396:
4391:
4378:
4373:
4360:
4355:
4342:
4329:
4328:
4316:
4315:
4303:
4302:
4290:
4289:
4277:
4271:
4251:
4247:
4244:
4239:
4234:
4221:
4216:
4203:
4198:
4185:
4180:
4167:
4163:
4162:
4150:
4149:
4137:
4136:
4124:
4123:
4111:
4105:
4075:
4073:
4072:
4067:
4065:
4035:
4031:
4028:
4023:
4018:
4005:
4000:
3987:
3982:
3969:
3964:
3951:
3947:
3946:
3934:
3933:
3921:
3920:
3908:
3907:
3895:
3889:
3851:
3849:
3848:
3843:
3828:
3826:
3825:
3820:
3818:
3817:
3798:
3796:
3795:
3790:
3782:
3781:
3772:
3770:
3762:
3752:
3751:
3735:
3730:
3729:
3720:
3718:
3710:
3700:
3699:
3683:
3675:
3673:
3665:
3655:
3654:
3638:
3593:
3591:
3590:
3585:
3583:
3582:
3570:
3569:
3548:
3547:
3532:
3531:
3519:
3518:
3487:
3485:
3484:
3479:
3474:
3473:
3461:
3430:
3428:
3427:
3422:
3417:
3416:
3404:
3365:
3363:
3362:
3357:
3352:
3351:
3339:
3310:Taylor expansion
3307:
3305:
3304:
3299:
3284:
3282:
3281:
3276:
3271:
3270:
3249:
3248:
3233:
3232:
3220:
3219:
3181:
3179:
3178:
3173:
3168:
3167:
3088:
3086:
3085:
3080:
3068:
3066:
3065:
3060:
3018:
3016:
3015:
3010:
3008:
3004:
3003:
3002:
2997:
2991:
2988:
2983:
2980:
2975:
2917:
2915:
2914:
2909:
2907:
2904:
2899:
2896:
2878:
2876:
2875:
2870:
2868:
2867:
2862:
2849:
2847:
2846:
2841:
2824:
2822:
2821:
2816:
2792:
2790:
2789:
2784:
2782:
2779:
2774:
2773:
2768:
2765:
2743:over a field. A
2735:
2733:
2732:
2727:
2725:
2724:
2719:
2716:
2691:
2689:
2688:
2683:
2681:
2680:
2664:
2662:
2661:
2656:
2651:
2650:
2638:
2637:
2625:
2624:
2612:
2611:
2596:
2595:
2583:
2582:
2548:
2546:
2545:
2540:
2538:
2537:
2525:
2524:
2484:
2482:
2481:
2476:
2474:
2472:
2468:
2467:
2448:
2447:
2438:
2426:
2424:
2423:
2418:
2416:
2415:
2403:
2402:
2383:
2381:
2380:
2375:
2373:
2371:
2367:
2366:
2364:
2363:
2362:
2349:
2341:
2330:
2328:
2327:
2326:
2313:
2305:
2298:
2297:
2288:
2274:
2273:
2254:
2252:
2251:
2246:
2234:
2232:
2231:
2226:
2224:
2223:
2218:
2209:
2208:
2203:
2191:
2190:
2172:
2171:
2146:
2144:
2143:
2138:
2120:
2118:
2117:
2112:
2110:
2108:
2107:
2106:
2087:
2086:
2077:
2063:
2062:
2040:
2038:
2037:
2032:
2030:
2028:
2024:
2023:
2005:
2004:
1991:
1987:
1986:
1968:
1967:
1955:
1949:
1937:
1935:
1934:
1929:
1914:
1912:
1911:
1906:
1889:
1888:
1861:
1860:
1859:
1858:
1848:
1845:
1833:
1829:
1828:
1816:
1813:
1799:
1798:
1773:
1771:
1770:
1765:
1753:
1751:
1750:
1745:
1734:
1733:
1717:
1715:
1714:
1709:
1698:
1697:
1681:
1679:
1678:
1673:
1657:
1655:
1654:
1649:
1635:
1634:
1622:
1619:
1604:
1602:
1601:
1596:
1576:
1575:
1574:
1556:
1555:
1554:
1533:
1532:
1531:
1510:
1496:
1494:
1493:
1488:
1480:
1479:
1463:
1461:
1460:
1455:
1444:
1443:
1416:
1414:
1413:
1408:
1379:
1377:
1376:
1371:
1356:
1354:
1353:
1348:
1346:
1333:
1321:
1308:
1299:
1297:
1287:
1286:
1274:
1273:
1260:
1238:
1232:
1223:
1221:
1217:
1216:
1204:
1203:
1190:
1174:
1168:
1149:
1147:
1146:
1141:
1139:
1125:
1109:
1090:
1065:
1063:
1062:
1057:
1055:
1046:
1042:
1018:
999:
998:
975:
973:
972:
967:
965:
964:
948:
946:
945:
940:
928:
926:
925:
920:
908:
906:
905:
900:
876:
874:
873:
868:
866:
865:
838:
836:
835:
830:
828:
826:
822:
821:
809:
808:
795:
791:
790:
772:
771:
759:
753:
735:
733:
732:
727:
715:
713:
712:
707:
702:
701:
683:
682:
670:
655:
653:
652:
647:
645:
640:
636:
635:
617:
616:
604:
598:
551:moduli of curves
537:
535:
534:
529:
527:
502:
493:
492:
456:
454:
453:
448:
440:
439:
426:universal family
419:
417:
416:
411:
338:cotangent bundle
332:
330:
329:
324:
319:
318:
300:
299:
196:
194:
193:
188:
176:
175:
155:sheaf cohomology
139:Riemann surfaces
116:Kunihiko Kodaira
27:is the study of
7839:
7838:
7834:
7833:
7832:
7830:
7829:
7828:
7809:
7808:
7800:
7796:
7793:
7783:
7776:
7742:
7734:
7731:
7729:Survey articles
7709:
7701:
7677:
7663:(Google eBook)
7655:, eds. (1992).
7630:
7627:
7622:
7621:
7608:
7607:
7603:
7596:
7575:
7574:
7565:
7560:
7513:
7489:
7453:
7437:
7426:
7425:
7393:
7377:
7366:
7365:
7356:
7350:
7317:
7306:
7305:
7285:
7257:
7252:
7251:
7223:
7199:
7198:
7178:
7177:
7169:
7153:
7129:
7124:
7108:
7084:
7079:
7064:
7052:
7050:
7044:
7038:
7032:
7031:
7023:
7009:
7007:
7002:
6988:
6986:
6981:
6969:
6968:
6948:
6947:
6931:
6907:
6902:
6887:
6875:
6874:
6868:
6862:
6861:
6847:
6845:
6840:
6828:
6827:
6807:
6806:
6791:
6779:
6778:
6772:
6771:
6759:
6758:
6732:
6727:
6726:
6705:
6700:
6699:
6676:
6659:
6658:
6635:
6622:
6621:
6618:
6566:rational curves
6554:
6521:
6520:
6484:
6479:
6478:
6461:
6460:
6439:
6438:
6404:
6391:
6378:
6340:
6339:
6309:
6296:
6256:
6213:
6208:
6207:
6147:
6142:
6141:
6116:
6115:
6098:
6097:
6076:
6075:
6035:
5995:
5952:
5943:
5942:
5918:
5917:
5907:
5870:
5861:
5860:
5850:
5840:
5803:
5796:
5783:
5764:
5755:
5754:
5728:
5727:
5700:
5681:
5655:
5641:
5640:
5612:
5607:
5606:
5600:
5595:
5542:
5537:
5536:
5456:
5442:
5441:
5419:
5418:
5399:
5398:
5395:
5373:
5372:
5361:
5350:
5345:
5328:
5323:
5305:
5304:
5298:
5292:
5286:
5285:
5273:
5271:
5266:
5257:
5252:
5240:
5239:
5219:
5218:
5201:
5196:
5178:
5177:
5171:
5165:
5164:
5155:
5150:
5138:
5137:
5094:
5083:
5082:
5054:
5049:
5048:
5041:
5017:
5016:
4997:
4996:
4975:
4974:
4957:
4952:
4934:
4933:
4927:
4921:
4920:
4911:
4906:
4822:
4808:
4795:
4782:
4769:
4760:
4753:
4739:
4719:
4718:
4675:
4674:
4647:
4634:
4621:
4608:
4603:
4602:
4585:
4584:
4558:
4553:
4535:
4534:
4528:
4522:
4521:
4498:
4476:
4454:
4432:
4343:
4320:
4307:
4294:
4281:
4272:
4265:
4257:
4252:
4168:
4154:
4141:
4128:
4115:
4106:
4099:
4084:
4083:
4063:
4062:
4044:
4043:
4037:
4036:
3952:
3938:
3925:
3912:
3899:
3890:
3883:
3868:
3867:
3861:
3831:
3830:
3809:
3804:
3803:
3773:
3763:
3737:
3736:
3721:
3711:
3685:
3684:
3666:
3640:
3639:
3602:
3601:
3574:
3561:
3539:
3523:
3510:
3493:
3492:
3465:
3440:
3439:
3408:
3368:
3367:
3343:
3318:
3317:
3290:
3289:
3262:
3240:
3224:
3211:
3194:
3193:
3159:
3094:
3093:
3071:
3070:
3036:
3035:
3032:
2992:
2973:
2972:
2966:
2965:
2959:
2958:
2950:
2946:
2926:
2925:
2884:
2883:
2857:
2852:
2851:
2832:
2831:
2798:
2797:
2763:
2752:
2751:
2714:
2709:
2708:
2698:
2672:
2667:
2666:
2642:
2629:
2616:
2603:
2587:
2574:
2551:
2550:
2529:
2516:
2490:
2489:
2459:
2449:
2439:
2432:
2431:
2407:
2394:
2389:
2388:
2354:
2350:
2342:
2318:
2314:
2306:
2303:
2299:
2289:
2265:
2260:
2259:
2237:
2236:
2213:
2198:
2182:
2163:
2149:
2148:
2126:
2125:
2098:
2088:
2078:
2054:
2049:
2048:
2015:
1996:
1992:
1978:
1959:
1950:
1943:
1942:
1920:
1919:
1880:
1850:
1843:
1820:
1790:
1779:
1778:
1756:
1755:
1725:
1720:
1719:
1689:
1684:
1683:
1664:
1663:
1626:
1610:
1609:
1566:
1543:
1520:
1502:
1501:
1471:
1466:
1465:
1435:
1427:
1426:
1399:
1398:
1390:
1362:
1361:
1344:
1343:
1327:
1322:
1314:
1313:
1307:
1301:
1300:
1278:
1265:
1261:
1233:
1229:
1224:
1208:
1195:
1191:
1169:
1158:
1157:
1137:
1136:
1131:
1126:
1118:
1115:
1114:
1108:
1102:
1101:
1096:
1091:
1083:
1074:
1073:
1053:
1052:
1047:
1030:
1024:
1023:
1017:
1011:
1010:
1005:
1000:
990:
981:
980:
956:
951:
950:
931:
930:
911:
910:
879:
878:
857:
852:
851:
813:
800:
796:
782:
763:
754:
741:
740:
718:
717:
693:
674:
661:
660:
627:
608:
599:
586:
585:
567:Stein manifolds
563:
525:
524:
519:
514:
508:
507:
501:
495:
494:
485:
480:
468:
467:
462:pullback square
429:
428:
390:
389:
386:
304:
291:
286:
285:
277:to relate the
167:
162:
161:
137:In the case of
104:
41:
17:
12:
11:
5:
7837:
7835:
7827:
7826:
7821:
7811:
7810:
7807:
7806:
7792:
7791:External links
7789:
7788:
7787:
7774:
7755:(3): 307–336,
7730:
7727:
7726:
7725:
7720:
7715:
7707:
7699:
7694:
7689:
7676:
7673:
7672:
7671:
7646:
7626:
7623:
7620:
7619:
7601:
7594:
7562:
7561:
7559:
7556:
7555:
7554:
7549:
7544:
7539:
7534:
7529:
7524:
7519:
7512:
7509:
7491:The so-called
7488:
7485:
7484:
7483:
7467:
7462:
7457:
7452:
7449:
7444:
7440:
7436:
7433:
7419:
7418:
7407:
7402:
7397:
7392:
7389:
7384:
7380:
7376:
7373:
7349:
7346:
7329:
7324:
7320:
7316:
7313:
7293:abelian scheme
7284:
7281:
7266:
7261:
7239:
7234:
7228:
7222:
7219:
7216:
7213:
7208:
7192:
7191:
7176:
7173:
7170:
7168:
7165:
7160:
7156:
7152:
7148:
7143:
7139:
7136:
7133:
7130:
7128:
7125:
7123:
7120:
7115:
7111:
7107:
7103:
7098:
7094:
7091:
7088:
7085:
7083:
7080:
7078:
7073:
7068:
7063:
7060:
7057:
7054:
7053:
7049:
7046:
7043:
7040:
7037:
7034:
7033:
7030:
7027:
7024:
7020:
7014:
7008:
7006:
7003:
6999:
6993:
6987:
6985:
6982:
6980:
6977:
6976:
6962:
6961:
6946:
6943:
6938:
6934:
6930:
6926:
6921:
6917:
6914:
6911:
6908:
6906:
6903:
6901:
6896:
6891:
6886:
6883:
6880:
6877:
6876:
6873:
6870:
6867:
6864:
6863:
6858:
6852:
6846:
6844:
6841:
6839:
6836:
6835:
6821:
6820:
6805:
6800:
6795:
6790:
6787:
6784:
6781:
6780:
6777:
6774:
6773:
6770:
6767:
6766:
6741:
6736:
6712:
6708:
6685:
6680:
6674:
6668:
6644:
6639:
6633:
6629:
6617:
6614:
6562:Shigefumi Mori
6553:
6552:Bend-and-break
6550:
6537:
6534:
6531:
6528:
6508:
6505:
6502:
6499:
6496:
6491:
6487:
6459:
6456:
6453:
6450:
6447:
6444:
6442:
6440:
6437:
6434:
6431:
6428:
6425:
6422:
6419:
6416:
6413:
6410:
6407:
6405:
6403:
6398:
6394:
6390:
6385:
6381:
6375:
6372:
6367:
6363:
6359:
6356:
6348:
6347:
6321:
6316:
6312:
6308:
6303:
6299:
6293:
6290:
6285:
6281:
6277:
6274:
6271:
6268:
6263:
6259:
6255:
6252:
6247:
6244:
6239:
6235:
6231:
6228:
6225:
6220:
6216:
6192:
6189:
6186:
6181:
6178:
6173:
6169:
6165:
6162:
6159:
6154:
6150:
6129:
6126:
6123:
6096:
6093:
6090:
6087:
6084:
6081:
6079:
6077:
6074:
6071:
6068:
6065:
6062:
6059:
6056:
6053:
6050:
6047:
6044:
6041:
6038:
6036:
6034:
6029:
6026:
6021:
6017:
6013:
6010:
6007:
6002:
5998:
5994:
5991:
5986:
5983:
5978:
5974:
5970:
5967:
5964:
5959:
5955:
5951:
5950:
5914:
5910:
5904:
5901:
5896:
5892:
5888:
5885:
5882:
5877:
5873:
5869:
5866:
5864:
5862:
5857:
5853:
5847:
5843:
5839:
5834:
5829:
5825:
5821:
5818:
5815:
5810:
5806:
5802:
5799:
5797:
5795:
5790:
5786:
5782:
5779:
5776:
5771:
5767:
5763:
5762:
5735:
5712:
5707:
5703:
5699:
5696:
5693:
5688:
5684:
5680:
5677:
5674:
5671:
5666:
5660:
5654:
5651:
5648:
5623:
5617:
5599:
5596:
5594:
5591:
5590:
5589:
5578:
5575:
5572:
5569:
5566:
5563:
5560:
5557:
5554:
5549:
5545:
5526:
5525:
5514:
5511:
5508:
5505:
5502:
5499:
5496:
5493:
5490:
5487:
5484:
5481:
5478:
5473:
5469:
5459:
5455:
5452:
5449:
5426:
5406:
5394:
5391:
5387:
5386:
5371:
5367:
5364:
5360:
5357:
5354:
5351:
5349:
5346:
5344:
5341:
5338:
5335:
5332:
5329:
5327:
5324:
5322:
5319:
5316:
5313:
5310:
5307:
5306:
5303:
5300:
5297:
5294:
5291:
5288:
5287:
5283:
5278:
5272:
5270:
5267:
5263:
5258:
5256:
5253:
5251:
5248:
5247:
5233:
5232:
5217:
5214:
5211:
5208:
5205:
5202:
5200:
5197:
5195:
5192:
5189:
5186:
5183:
5180:
5179:
5176:
5173:
5170:
5167:
5166:
5161:
5156:
5154:
5151:
5149:
5146:
5145:
5131:
5130:
5119:
5116:
5113:
5110:
5107:
5104:
5100:
5097:
5093:
5090:
5067:
5064:
5060:
5057:
5040:
5037:
5024:
5004:
4993:
4992:
4980:
4973:
4970:
4967:
4964:
4961:
4958:
4956:
4953:
4951:
4948:
4945:
4942:
4939:
4936:
4935:
4932:
4929:
4926:
4923:
4922:
4917:
4912:
4910:
4907:
4904:
4897:
4892:
4887:
4883:
4879:
4874:
4869:
4865:
4861:
4856:
4851:
4847:
4843:
4838:
4833:
4829:
4825:
4820:
4815:
4811:
4807:
4802:
4798:
4794:
4789:
4785:
4781:
4776:
4772:
4768:
4764:
4756:
4752:
4749:
4746:
4745:
4742:
4738:
4735:
4732:
4729:
4726:
4703:
4700:
4697:
4694:
4691:
4688:
4685:
4682:
4662:
4659:
4654:
4650:
4646:
4641:
4637:
4633:
4628:
4624:
4620:
4615:
4611:
4599:
4598:
4583:
4580:
4577:
4574:
4571:
4568:
4565:
4562:
4559:
4557:
4554:
4552:
4549:
4546:
4543:
4540:
4537:
4536:
4533:
4530:
4527:
4524:
4523:
4519:
4512:
4505:
4501:
4495:
4491:
4483:
4479:
4473:
4469:
4461:
4457:
4451:
4447:
4439:
4435:
4429:
4425:
4421:
4418:
4413:
4408:
4404:
4400:
4395:
4390:
4386:
4382:
4377:
4372:
4368:
4364:
4359:
4354:
4350:
4346:
4341:
4338:
4335:
4332:
4327:
4323:
4319:
4314:
4310:
4306:
4301:
4297:
4293:
4288:
4284:
4280:
4276:
4268:
4264:
4261:
4258:
4256:
4253:
4250:
4243:
4238:
4233:
4229:
4225:
4220:
4215:
4211:
4207:
4202:
4197:
4193:
4189:
4184:
4179:
4175:
4171:
4166:
4161:
4157:
4153:
4148:
4144:
4140:
4135:
4131:
4127:
4122:
4118:
4114:
4110:
4102:
4098:
4095:
4092:
4091:
4077:
4076:
4061:
4058:
4055:
4052:
4049:
4046:
4045:
4042:
4039:
4038:
4034:
4027:
4022:
4017:
4013:
4009:
4004:
3999:
3995:
3991:
3986:
3981:
3977:
3973:
3968:
3963:
3959:
3955:
3950:
3945:
3941:
3937:
3932:
3928:
3924:
3919:
3915:
3911:
3906:
3902:
3898:
3894:
3886:
3882:
3879:
3876:
3875:
3860:
3857:
3841:
3838:
3816:
3812:
3800:
3799:
3788:
3785:
3780:
3776:
3769:
3766:
3761:
3758:
3755:
3750:
3747:
3744:
3740:
3733:
3728:
3724:
3717:
3714:
3709:
3706:
3703:
3698:
3695:
3692:
3688:
3681:
3678:
3672:
3669:
3664:
3661:
3658:
3653:
3650:
3647:
3643:
3636:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3595:
3594:
3581:
3577:
3573:
3568:
3564:
3560:
3557:
3554:
3551:
3546:
3542:
3538:
3535:
3530:
3526:
3522:
3517:
3513:
3509:
3506:
3503:
3500:
3477:
3472:
3468:
3464:
3460:
3456:
3453:
3450:
3447:
3420:
3415:
3411:
3407:
3403:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3355:
3350:
3346:
3342:
3338:
3334:
3331:
3328:
3325:
3297:
3286:
3285:
3274:
3269:
3265:
3261:
3258:
3255:
3252:
3247:
3243:
3239:
3236:
3231:
3227:
3223:
3218:
3214:
3210:
3207:
3204:
3201:
3183:
3182:
3171:
3166:
3162:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3078:
3058:
3055:
3052:
3049:
3046:
3043:
3031:
3028:
3020:
3019:
3007:
3001:
2996:
2986:
2978:
2971:
2968:
2967:
2964:
2961:
2960:
2957:
2954:
2953:
2949:
2945:
2942:
2939:
2936:
2933:
2919:
2918:
2902:
2894:
2891:
2866:
2861:
2839:
2814:
2811:
2808:
2805:
2794:
2793:
2777:
2772:
2762:
2759:
2741:Artin algebras
2723:
2697:
2694:
2679:
2675:
2654:
2649:
2645:
2641:
2636:
2632:
2628:
2623:
2619:
2615:
2610:
2606:
2602:
2599:
2594:
2590:
2586:
2581:
2577:
2573:
2570:
2567:
2564:
2561:
2558:
2536:
2532:
2528:
2523:
2519:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2471:
2466:
2462:
2458:
2455:
2452:
2446:
2442:
2414:
2410:
2406:
2401:
2397:
2370:
2361:
2357:
2353:
2348:
2345:
2339:
2336:
2333:
2325:
2321:
2317:
2312:
2309:
2302:
2296:
2292:
2286:
2283:
2280:
2277:
2272:
2268:
2244:
2222:
2217:
2212:
2207:
2202:
2197:
2194:
2189:
2185:
2181:
2178:
2175:
2170:
2166:
2162:
2159:
2156:
2136:
2133:
2105:
2101:
2097:
2094:
2091:
2085:
2081:
2075:
2072:
2069:
2066:
2061:
2057:
2027:
2022:
2018:
2014:
2011:
2008:
2003:
1999:
1995:
1990:
1985:
1981:
1977:
1974:
1971:
1966:
1962:
1958:
1954:
1927:
1904:
1901:
1898:
1895:
1892:
1887:
1883:
1879:
1876:
1873:
1870:
1867:
1864:
1857:
1853:
1840:
1836:
1832:
1827:
1823:
1819:
1811:
1808:
1805:
1802:
1797:
1793:
1789:
1786:
1763:
1743:
1740:
1737:
1732:
1728:
1707:
1704:
1701:
1696:
1692:
1671:
1647:
1644:
1641:
1638:
1633:
1629:
1625:
1617:
1594:
1591:
1588:
1583:
1579:
1573:
1569:
1563:
1559:
1553:
1550:
1546:
1540:
1536:
1530:
1527:
1523:
1517:
1513:
1509:
1486:
1483:
1478:
1474:
1453:
1450:
1447:
1442:
1438:
1434:
1423:Galina Tyurina
1406:
1389:
1386:
1380:is called the
1369:
1342:
1339:
1336:
1332:
1328:
1326:
1323:
1320:
1316:
1315:
1312:
1309:
1306:
1303:
1302:
1296:
1293:
1290:
1285:
1281:
1277:
1272:
1268:
1264:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1237:
1230:
1228:
1225:
1220:
1215:
1211:
1207:
1202:
1198:
1194:
1189:
1186:
1183:
1180:
1177:
1173:
1166:
1165:
1135:
1132:
1130:
1127:
1124:
1121:
1117:
1116:
1113:
1110:
1107:
1104:
1103:
1100:
1097:
1095:
1092:
1089:
1086:
1082:
1081:
1051:
1048:
1041:
1036:
1031:
1029:
1026:
1025:
1022:
1019:
1016:
1013:
1012:
1009:
1006:
1004:
1001:
997:
993:
989:
988:
963:
959:
949:such that the
938:
918:
898:
895:
892:
889:
886:
864:
860:
825:
820:
816:
812:
807:
803:
799:
794:
789:
785:
781:
778:
775:
770:
766:
762:
758:
751:
748:
725:
705:
700:
696:
692:
689:
686:
681:
677:
673:
669:
643:
639:
634:
630:
626:
623:
620:
615:
611:
607:
603:
596:
593:
562:
559:
543:Hilbert scheme
523:
520:
518:
515:
513:
510:
509:
506:
503:
500:
497:
496:
491:
486:
484:
481:
479:
476:
475:
446:
443:
438:
409:
406:
403:
400:
397:
385:
382:
350:exterior power
334:
333:
322:
317:
314:
311:
307:
303:
298:
294:
273:> 1, using
206:tangent bundle
198:
197:
185:
182:
179:
174:
170:
147:elliptic curve
143:Riemann sphere
103:
100:
39:
15:
13:
10:
9:
6:
4:
3:
2:
7836:
7825:
7822:
7820:
7817:
7816:
7814:
7799:
7795:
7794:
7790:
7782:
7781:
7775:
7772:
7768:
7763:
7758:
7754:
7750:
7749:
7741:
7737:
7733:
7732:
7728:
7724:
7721:
7719:
7716:
7713:
7708:
7705:
7700:
7698:
7695:
7693:
7690:
7687:
7683:
7679:
7678:
7674:
7670:
7666:
7662:
7658:
7654:
7650:
7647:
7643:
7639:
7638:
7633:
7632:"deformation"
7629:
7628:
7624:
7615:
7611:
7605:
7602:
7597:
7591:
7587:
7583:
7579:
7572:
7570:
7568:
7564:
7557:
7553:
7550:
7548:
7545:
7543:
7540:
7538:
7535:
7533:
7530:
7528:
7525:
7523:
7520:
7518:
7515:
7514:
7510:
7508:
7506:
7502:
7501:string theory
7498:
7494:
7486:
7460:
7447:
7442:
7438:
7431:
7424:
7423:
7422:
7400:
7387:
7382:
7378:
7371:
7364:
7363:
7362:
7361:
7355:
7347:
7345:
7343:
7318:
7311:
7304:
7302:
7297:
7294:
7290:
7282:
7280:
7264:
7232:
7217:
7214:
7211:
7197:
7196:formal scheme
7174:
7158:
7154:
7146:
7134:
7131:
7113:
7109:
7101:
7089:
7086:
7071:
7058:
7055:
7028:
7018:
6997:
6978:
6967:
6966:
6965:
6936:
6932:
6924:
6912:
6909:
6894:
6881:
6878:
6856:
6837:
6826:
6825:
6824:
6798:
6785:
6782:
6768:
6757:
6756:
6755:
6739:
6710:
6706:
6683:
6672:
6642:
6631:
6627:
6615:
6613:
6611:
6607:
6603:
6599:
6595:
6591:
6587:
6583:
6579:
6575:
6571:
6567:
6563:
6559:
6551:
6549:
6535:
6532:
6529:
6526:
6506:
6503:
6497:
6489:
6485:
6474:
6457:
6454:
6451:
6448:
6445:
6443:
6435:
6432:
6429:
6426:
6423:
6420:
6417:
6414:
6411:
6408:
6406:
6396:
6392:
6388:
6383:
6373:
6370:
6365:
6361:
6336:
6335:the degree is
6332:
6314:
6310:
6306:
6301:
6291:
6288:
6283:
6279:
6272:
6269:
6261:
6257:
6253:
6245:
6242:
6237:
6233:
6229:
6226:
6218:
6214:
6204:
6190:
6187:
6179:
6176:
6171:
6167:
6163:
6160:
6152:
6148:
6127:
6124:
6121:
6111:
6094:
6091:
6088:
6085:
6082:
6080:
6072:
6069:
6066:
6063:
6057:
6054:
6051:
6048:
6042:
6039:
6037:
6027:
6024:
6019:
6015:
6011:
6008:
6000:
5996:
5992:
5984:
5981:
5976:
5972:
5968:
5965:
5957:
5953:
5939:
5937:
5931:
5912:
5902:
5899:
5894:
5890:
5886:
5883:
5875:
5871:
5867:
5865:
5855:
5845:
5841:
5837:
5832:
5827:
5823:
5819:
5816:
5808:
5804:
5800:
5798:
5788:
5784:
5780:
5777:
5769:
5765:
5751:
5749:
5748:Serre duality
5733:
5723:
5705:
5701:
5697:
5694:
5686:
5682:
5678:
5675:
5672:
5664:
5649:
5646:
5637:
5621:
5605:
5597:
5592:
5576:
5567:
5561:
5555:
5552:
5547:
5543:
5535:
5534:
5533:
5531:
5509:
5506:
5497:
5491:
5485:
5482:
5476:
5471:
5467:
5457:
5453:
5450:
5447:
5440:
5439:
5438:
5424:
5404:
5393:Tangent space
5392:
5390:
5365:
5362:
5355:
5352:
5339:
5333:
5330:
5317:
5311:
5308:
5281:
5249:
5238:
5237:
5236:
5212:
5206:
5203:
5190:
5184:
5181:
5147:
5136:
5135:
5134:
5114:
5108:
5098:
5095:
5088:
5081:
5080:
5079:
5065:
5058:
5055:
5046:
5038:
5036:
5022:
5002:
4978:
4968:
4962:
4959:
4946:
4940:
4937:
4902:
4890:
4885:
4881:
4877:
4872:
4867:
4863:
4859:
4854:
4849:
4845:
4841:
4836:
4831:
4827:
4813:
4809:
4805:
4800:
4796:
4792:
4787:
4783:
4779:
4774:
4770:
4754:
4750:
4747:
4740:
4736:
4730:
4724:
4717:
4716:
4715:
4695:
4689:
4683:
4680:
4660:
4657:
4652:
4648:
4644:
4639:
4635:
4631:
4626:
4622:
4618:
4613:
4609:
4575:
4569:
4563:
4560:
4547:
4541:
4538:
4517:
4503:
4499:
4493:
4489:
4481:
4477:
4471:
4467:
4459:
4455:
4449:
4445:
4437:
4433:
4427:
4423:
4419:
4416:
4411:
4406:
4402:
4398:
4393:
4388:
4384:
4380:
4375:
4370:
4366:
4362:
4357:
4352:
4348:
4336:
4325:
4321:
4317:
4312:
4308:
4304:
4299:
4295:
4291:
4286:
4282:
4266:
4262:
4259:
4248:
4236:
4231:
4227:
4223:
4218:
4213:
4209:
4205:
4200:
4195:
4191:
4187:
4182:
4177:
4173:
4159:
4155:
4151:
4146:
4142:
4138:
4133:
4129:
4125:
4120:
4116:
4100:
4096:
4093:
4082:
4081:
4080:
4056:
4050:
4047:
4032:
4020:
4015:
4011:
4007:
4002:
3997:
3993:
3989:
3984:
3979:
3975:
3971:
3966:
3961:
3957:
3943:
3939:
3935:
3930:
3926:
3922:
3917:
3913:
3909:
3904:
3900:
3884:
3880:
3877:
3866:
3865:
3864:
3858:
3856:
3853:
3839:
3836:
3814:
3810:
3786:
3783:
3778:
3774:
3767:
3764:
3756:
3745:
3738:
3731:
3726:
3722:
3715:
3712:
3704:
3693:
3686:
3679:
3676:
3670:
3667:
3659:
3648:
3641:
3634:
3628:
3622:
3619:
3613:
3607:
3600:
3599:
3598:
3579:
3575:
3571:
3566:
3562:
3558:
3555:
3552:
3549:
3544:
3540:
3536:
3533:
3528:
3524:
3520:
3515:
3507:
3504:
3501:
3491:
3490:
3489:
3470:
3466:
3458:
3451:
3445:
3436:
3434:
3413:
3409:
3401:
3394:
3388:
3385:
3379:
3373:
3348:
3344:
3336:
3329:
3323:
3315:
3311:
3295:
3267:
3263:
3256:
3253:
3250:
3245:
3241:
3237:
3234:
3229:
3225:
3221:
3216:
3208:
3205:
3202:
3192:
3191:
3190:
3188:
3164:
3160:
3153:
3150:
3144:
3138:
3135:
3132:
3126:
3120:
3117:
3111:
3108:
3105:
3099:
3092:
3091:
3090:
3076:
3053:
3050:
3047:
3041:
3029:
3027:
3025:
3005:
2999:
2984:
2976:
2969:
2955:
2947:
2943:
2937:
2931:
2924:
2923:
2922:
2892:
2889:
2882:
2881:
2880:
2864:
2837:
2828:
2809:
2803:
2770:
2760:
2757:
2750:
2749:
2748:
2746:
2742:
2739:
2721:
2707:
2703:
2695:
2693:
2677:
2673:
2652:
2647:
2643:
2639:
2634:
2630:
2626:
2621:
2617:
2613:
2608:
2604:
2600:
2592:
2588:
2584:
2579:
2575:
2571:
2568:
2565:
2562:
2556:
2534:
2530:
2526:
2521:
2517:
2513:
2507:
2504:
2501:
2495:
2485:
2464:
2460:
2456:
2453:
2444:
2440:
2428:
2412:
2408:
2404:
2399:
2395:
2384:
2368:
2359:
2355:
2346:
2337:
2334:
2331:
2323:
2319:
2310:
2300:
2294:
2290:
2284:
2278:
2270:
2266:
2256:
2242:
2220:
2205:
2195:
2187:
2183:
2179:
2176:
2173:
2168:
2164:
2157:
2154:
2134:
2131:
2121:
2103:
2099:
2095:
2092:
2089:
2083:
2079:
2073:
2067:
2059:
2055:
2045:
2041:
2020:
2016:
2012:
2009:
2006:
2001:
1997:
1983:
1979:
1975:
1972:
1969:
1964:
1960:
1939:
1925:
1915:
1902:
1893:
1885:
1881:
1871:
1868:
1865:
1855:
1851:
1838:
1834:
1825:
1821:
1803:
1795:
1791:
1784:
1775:
1761:
1738:
1730:
1726:
1702:
1694:
1690:
1669:
1661:
1642:
1639:
1631:
1627:
1605:
1592:
1586:
1581:
1577:
1571:
1567:
1561:
1557:
1551:
1548:
1544:
1538:
1534:
1528:
1525:
1521:
1515:
1511:
1507:
1498:
1484:
1476:
1472:
1448:
1445:
1440:
1436:
1424:
1420:
1404:
1396:
1387:
1385:
1383:
1367:
1357:
1337:
1291:
1288:
1283:
1279:
1275:
1270:
1266:
1254:
1251:
1248:
1245:
1242:
1213:
1209:
1205:
1200:
1196:
1184:
1181:
1178:
1154:
1150:
1133:
1122:
1119:
1098:
1087:
1084:
1070:
1066:
1049:
1039:
1034:
1027:
1007:
995:
991:
977:
961:
957:
936:
916:
896:
890:
887:
884:
862:
858:
849:
845:
839:
818:
814:
810:
805:
801:
787:
783:
779:
776:
773:
768:
764:
749:
746:
737:
723:
698:
694:
690:
687:
684:
679:
675:
656:
641:
632:
628:
624:
621:
618:
613:
609:
594:
591:
582:
580:
576:
572:
568:
560:
558:
556:
552:
548:
544:
538:
521:
511:
477:
464:
463:
460:
444:
427:
423:
407:
401:
398:
395:
383:
381:
379:
375:
369:
367:
363:
359:
355:
351:
347:
343:
342:tensor square
339:
312:
296:
292:
284:
283:
282:
280:
276:
275:Serre duality
272:
267:
265:
261:
257:
253:
249:
245:
242:
238:
234:
230:
226:
222:
219:
215:
211:
207:
203:
172:
168:
160:
159:
158:
156:
152:
148:
144:
140:
135:
133:
129:
125:
121:
117:
113:
109:
101:
99:
97:
93:
89:
85:
81:
77:
73:
69:
64:
60:
55:
53:
49:
45:
38:
34:
30:
29:infinitesimal
26:
22:
7779:
7752:
7746:
7736:Mazur, Barry
7711:
7703:
7685:
7656:
7635:
7613:
7604:
7577:
7490:
7420:
7357:
7341:
7300:
7295:
7286:
7193:
6963:
6822:
6619:
6605:
6601:
6589:
6588:. Replacing
6581:
6577:
6574:Fano variety
6555:
6476:
6338:
6334:
6206:
6113:
5941:
5933:
5753:
5725:
5639:
5601:
5530:dual numbers
5527:
5396:
5388:
5234:
5132:
5044:
5042:
4994:
4600:
4078:
3862:
3854:
3801:
3596:
3437:
3287:
3184:
3033:
3021:
2920:
2827:moduli space
2795:
2744:
2699:
2487:
2430:
2386:
2258:
2123:
2047:
2043:
1941:
1917:
1777:
1659:
1607:
1500:
1418:
1391:
1382:Milnor fiber
1381:
1359:
1156:
1152:
1072:
1068:
979:
847:
843:
841:
739:
658:
584:
578:
564:
554:
540:
466:
458:
425:
387:
378:Grothendieck
370:
361:
345:
341:
335:
278:
270:
268:
263:
259:
255:
251:
247:
243:
240:
236:
232:
228:
224:
220:
218:Hodge number
213:
209:
199:
136:
132:moduli space
105:
88:group action
83:
79:
62:
58:
56:
36:
32:
24:
18:
7675:Pedagogical
7522:Dual number
848:deformation
547:Quot scheme
348:the second
72:engineering
48:constraints
21:mathematics
7813:Categories
7777:Anel, M.,
7669:0821851411
6586:components
5934:Hence the
5035:-algebra.
3859:Motivation
2796:such that
2665:where the
1464:such that
579:globalized
84:open orbit
7642:EMS Press
7448:
7435:→
7388:
7375:→
7323:∞
7233:∙
7218:
7175:⋯
7172:→
7135:
7127:→
7090:
7082:→
7059:
7048:↓
7042:↓
7036:↓
7029:⋯
7026:→
7005:→
6984:→
6913:
6905:→
6882:
6872:↓
6866:↓
6843:→
6786:
6776:↓
6570:varieties
6533:−
6452:−
6433:−
6415:−
6393:ω
6389:⊗
6384:∨
6371:⊗
6362:ω
6311:ω
6307:⊗
6302:∨
6289:⊗
6280:ω
6243:⊗
6234:ω
6177:⊗
6168:ω
6125:≥
6092:−
6064:−
6055:−
6025:⊗
6016:ω
5993:−
5982:⊗
5973:ω
5913:∨
5900:⊗
5891:ω
5868:≅
5856:∨
5842:ω
5838:⊗
5833:∗
5801:≅
5679:
5650:
5568:ε
5498:ε
5486:
5477:
5356:
5348:→
5334:
5326:→
5312:
5302:↓
5296:↓
5290:↓
5269:→
5255:→
5207:
5199:→
5185:
5175:↓
5169:↓
5153:→
5106:→
5063:→
4963:
4955:→
4941:
4931:↓
4925:↓
4909:→
4751:
4696:ε
4684:
4576:ε
4564:
4556:→
4542:
4532:↓
4526:↓
4420:ε
4337:ε
4263:
4255:→
4097:
4051:
4041:↓
3881:
3787:⋯
3576:ε
3563:ε
3550:ε
3508:ε
3380:ε
3296:ε
3264:ε
3251:ε
3209:ε
3187:monomials
3161:ε
3136:ε
3118:≡
3112:ε
3077:ε
3054:ε
3024:groupoids
2963:↓
2901:→
2776:→
2614:−
2527:−
2405:−
2352:∂
2344:∂
2335:…
2316:∂
2308:∂
2285:≅
2211:→
2177:…
2096:⋅
2074:≅
2010:…
1973:…
1900:→
1878:→
1810:→
1788:→
1632:∙
1590:→
1549:−
1526:−
1508:⋯
1482:→
1441:∙
1325:←
1311:↑
1305:↑
1276:−
1227:←
1206:−
1129:→
1112:↓
1106:↓
1094:→
1028:∗
1021:↓
1015:↓
1003:→
894:→
811:−
777:…
750:≅
688:…
622:…
595:≅
517:→
505:↓
499:↓
483:→
442:→
405:→
306:Ω
181:Θ
96:operators
7738:(2004),
7532:Exalcomm
7511:See also
6572:. For a
5366:′
5282:′
5099:′
5059:′
2706:category
2702:functors
1835:→
1578:→
1558:→
1535:→
1512:→
1123:′
1088:′
1035:→
7771:2058289
7644:, 2001
7625:Sources
6596:or the
6203:because
2704:on the
422:schemes
130:with a
68:physics
7769:
7667:
7592:
6598:degree
5045:smooth
4995:where
4601:where
2921:where
1718:. The
909:where
659:where
555:versal
459:unique
157:group
86:(of a
52:deform
7801:(PDF)
7784:(PDF)
7743:(PDF)
7558:Notes
6594:genus
5938:gives
5437:-set
2738:local
2124:were
573:, or
202:germs
7665:ISBN
7651:and
7590:ISBN
7287:The
7215:Spet
7132:Spec
7087:Spec
7056:Spec
6910:Spec
6879:Spec
6783:Spec
6477:and
6140:the
5483:Spec
5353:Spec
5331:Spec
5309:Spec
5204:Spec
5182:Spec
4960:Spec
4938:Spec
4748:Proj
4681:Spec
4561:Spec
4539:Spec
4260:Proj
4094:Proj
4048:Spec
3878:Proj
3288:the
2905:Sets
2780:Sets
118:and
110:and
70:and
7757:doi
7582:doi
6600:of
6568:on
6560:by
6351:deg
5676:dim
5647:dim
5463:Sch
5458:Hom
5425:Hom
3829:is
3435:.
2897:Sch
2850:in
2766:Art
2736:of
2717:Art
2549:is
1918:If
1846:Hom
1814:Der
1620:Der
545:or
346:not
281:to
264:a,b
19:In
7815::
7767:MR
7765:,
7753:41
7751:,
7745:,
7684:.
7659:,
7640:,
7634:,
7588:.
7566:^
7439:GL
7379:GL
7279:.
6612:.
6548:.
5553::=
5454::=
3852:.
1384:.
569:,
557:.
368:.
258:+
256:ax
254:+
250:=
235:+
233:ax
231:+
227:=
98:.
63:or
23:,
7803:.
7759::
7598:.
7584::
7470:?
7466:)
7461:p
7456:Z
7451:(
7443:n
7432:G
7406:)
7401:p
7396:F
7391:(
7383:n
7372:G
7342:p
7328:]
7319:p
7315:[
7312:A
7301:p
7296:A
7265:p
7260:Z
7238:)
7227:X
7221:(
7212:=
7207:X
7167:)
7164:)
7159:3
7155:p
7151:(
7147:/
7142:Z
7138:(
7122:)
7119:)
7114:2
7110:p
7106:(
7102:/
7097:Z
7093:(
7077:)
7072:p
7067:F
7062:(
7019:3
7013:X
6998:2
6992:X
6979:X
6945:)
6942:)
6937:2
6933:p
6929:(
6925:/
6920:Z
6916:(
6900:)
6895:p
6890:F
6885:(
6857:2
6851:X
6838:X
6804:)
6799:p
6794:F
6789:(
6769:X
6740:p
6735:Z
6711:2
6707:H
6684:p
6679:Z
6673:/
6667:X
6643:p
6638:F
6632:/
6628:X
6606:C
6602:C
6590:C
6582:C
6536:3
6530:g
6527:3
6507:0
6504:=
6501:)
6498:L
6495:(
6490:0
6486:h
6458:g
6455:2
6449:2
6446:=
6436:2
6430:g
6427:2
6424:+
6421:g
6418:4
6412:4
6409:=
6402:)
6397:C
6380:)
6374:2
6366:C
6358:(
6355:(
6320:)
6315:C
6298:)
6292:2
6284:C
6276:(
6273:,
6270:C
6267:(
6262:0
6258:h
6254:=
6251:)
6246:2
6238:C
6230:,
6227:C
6224:(
6219:1
6215:h
6191:0
6188:=
6185:)
6180:2
6172:C
6164:,
6161:C
6158:(
6153:1
6149:h
6128:2
6122:g
6095:3
6089:g
6086:3
6083:=
6073:1
6070:+
6067:g
6061:)
6058:2
6052:g
6049:2
6046:(
6043:2
6040:=
6033:)
6028:2
6020:C
6012:,
6009:C
6006:(
6001:1
5997:h
5990:)
5985:2
5977:C
5969:,
5966:C
5963:(
5958:0
5954:h
5909:)
5903:2
5895:C
5887:,
5884:C
5881:(
5876:0
5872:H
5852:)
5846:C
5828:C
5824:T
5820:,
5817:C
5814:(
5809:0
5805:H
5794:)
5789:C
5785:T
5781:,
5778:C
5775:(
5770:1
5766:H
5734:g
5711:)
5706:C
5702:T
5698:,
5695:C
5692:(
5687:1
5683:H
5673:=
5670:)
5665:g
5659:M
5653:(
5622:g
5616:M
5577:.
5574:)
5571:]
5565:[
5562:k
5559:(
5556:F
5548:F
5544:T
5513:)
5510:X
5507:,
5504:)
5501:]
5495:[
5492:k
5489:(
5480:(
5472:k
5468:/
5451:X
5448:T
5405:X
5370:)
5363:A
5359:(
5343:)
5340:A
5337:(
5321:)
5318:k
5315:(
5277:X
5262:X
5250:X
5216:)
5213:A
5210:(
5194:)
5191:k
5188:(
5160:X
5148:X
5118:)
5115:A
5112:(
5109:F
5103:)
5096:A
5092:(
5089:F
5066:A
5056:A
5023:k
5003:A
4979:}
4972:)
4969:A
4966:(
4950:)
4947:k
4944:(
4916:X
4903:)
4896:)
4891:4
4886:3
4882:x
4878:+
4873:4
4868:2
4864:x
4860:+
4855:4
4850:1
4846:x
4842:+
4837:4
4832:0
4828:x
4824:(
4819:]
4814:3
4810:x
4806:,
4801:2
4797:x
4793:,
4788:1
4784:x
4780:,
4775:0
4771:x
4767:[
4763:C
4755:(
4741:{
4737:=
4734:)
4731:A
4728:(
4725:F
4702:)
4699:]
4693:[
4690:k
4687:(
4661:4
4658:=
4653:3
4649:a
4645:+
4640:2
4636:a
4632:+
4627:1
4623:a
4619:+
4614:0
4610:a
4582:)
4579:]
4573:[
4570:k
4567:(
4551:)
4548:k
4545:(
4518:)
4511:)
4504:3
4500:a
4494:3
4490:x
4482:2
4478:a
4472:2
4468:x
4460:1
4456:a
4450:1
4446:x
4438:0
4434:a
4428:0
4424:x
4417:+
4412:4
4407:3
4403:x
4399:+
4394:4
4389:2
4385:x
4381:+
4376:4
4371:1
4367:x
4363:+
4358:4
4353:0
4349:x
4345:(
4340:]
4334:[
4331:]
4326:3
4322:x
4318:,
4313:2
4309:x
4305:,
4300:1
4296:x
4292:,
4287:0
4283:x
4279:[
4275:C
4267:(
4249:)
4242:)
4237:4
4232:3
4228:x
4224:+
4219:4
4214:2
4210:x
4206:+
4201:4
4196:1
4192:x
4188:+
4183:4
4178:0
4174:x
4170:(
4165:]
4160:3
4156:x
4152:,
4147:2
4143:x
4139:,
4134:1
4130:x
4126:,
4121:0
4117:x
4113:[
4109:C
4101:(
4060:)
4057:k
4054:(
4033:)
4026:)
4021:4
4016:3
4012:x
4008:+
4003:4
3998:2
3994:x
3990:+
3985:4
3980:1
3976:x
3972:+
3967:4
3962:0
3958:x
3954:(
3949:]
3944:3
3940:x
3936:,
3931:2
3927:x
3923:,
3918:1
3914:x
3910:,
3905:0
3901:x
3897:[
3893:C
3885:(
3840:x
3837:6
3815:3
3811:x
3784:+
3779:3
3775:x
3768:!
3765:3
3760:)
3757:0
3754:(
3749:)
3746:3
3743:(
3739:f
3732:+
3727:2
3723:x
3716:!
3713:2
3708:)
3705:0
3702:(
3697:)
3694:2
3691:(
3687:f
3680:+
3677:x
3671:!
3668:1
3663:)
3660:0
3657:(
3652:)
3649:1
3646:(
3642:f
3635:+
3632:)
3629:0
3626:(
3623:f
3620:=
3617:)
3614:x
3611:(
3608:f
3580:3
3572:+
3567:2
3559:x
3556:3
3553:+
3545:2
3541:x
3537:3
3534:+
3529:3
3525:x
3521:=
3516:3
3512:)
3505:+
3502:x
3499:(
3476:)
3471:k
3467:y
3463:(
3459:/
3455:]
3452:y
3449:[
3446:k
3419:)
3414:2
3410:y
3406:(
3402:/
3398:]
3395:y
3392:[
3389:k
3386:=
3383:]
3377:[
3374:k
3354:)
3349:2
3345:y
3341:(
3337:/
3333:]
3330:y
3327:[
3324:k
3273:)
3268:2
3260:(
3257:O
3254:+
3246:2
3242:x
3238:3
3235:+
3230:3
3226:x
3222:=
3217:3
3213:)
3206:+
3203:x
3200:(
3170:)
3165:2
3157:(
3154:O
3151:+
3148:)
3145:x
3142:(
3139:g
3133:+
3130:)
3127:x
3124:(
3121:f
3115:)
3109:,
3106:x
3103:(
3100:F
3057:)
3051:,
3048:x
3045:(
3042:F
3006:}
3000:n
2995:P
2985:d
2977::
2970:S
2956:X
2948:{
2944:=
2941:)
2938:S
2935:(
2932:F
2893::
2890:F
2865:n
2860:P
2838:d
2813:)
2810:k
2807:(
2804:F
2771:k
2761::
2758:F
2722:k
2678:i
2674:a
2653:x
2648:2
2644:a
2640:+
2635:1
2631:a
2627:+
2622:3
2618:x
2609:2
2605:y
2601:=
2598:)
2593:2
2589:a
2585:,
2580:1
2576:a
2572:,
2569:y
2566:,
2563:x
2560:(
2557:F
2535:3
2531:x
2522:2
2518:y
2514:=
2511:)
2508:y
2505:,
2502:x
2499:(
2496:f
2470:)
2465:2
2461:x
2457:,
2454:y
2451:(
2445:2
2441:A
2413:3
2409:x
2400:2
2396:y
2369:)
2360:n
2356:z
2347:f
2338:,
2332:,
2324:1
2320:z
2311:f
2301:(
2295:n
2291:A
2282:)
2279:A
2276:(
2271:1
2267:T
2243:f
2221:m
2216:C
2206:n
2201:C
2196::
2193:)
2188:m
2184:f
2180:,
2174:,
2169:1
2165:f
2161:(
2158:=
2155:f
2135:f
2132:d
2104:n
2100:A
2093:f
2090:d
2084:m
2080:A
2071:)
2068:A
2065:(
2060:1
2056:T
2026:)
2021:m
2017:f
2013:,
2007:,
2002:1
1998:f
1994:(
1989:}
1984:n
1980:z
1976:,
1970:,
1965:1
1961:z
1957:{
1953:C
1926:A
1903:0
1897:)
1894:A
1891:(
1886:1
1882:T
1875:)
1872:A
1869:,
1866:I
1863:(
1856:0
1852:R
1839:d
1831:)
1826:0
1822:R
1818:(
1807:)
1804:A
1801:(
1796:0
1792:T
1785:0
1762:A
1742:)
1739:A
1736:(
1731:1
1727:T
1706:)
1703:A
1700:(
1695:k
1691:T
1670:A
1646:)
1643:d
1640:,
1637:)
1628:R
1624:(
1616:(
1593:0
1587:A
1582:p
1572:0
1568:R
1562:s
1552:1
1545:R
1539:s
1529:2
1522:R
1516:s
1485:A
1477:0
1473:R
1452:)
1449:s
1446:,
1437:R
1433:(
1405:A
1368:s
1341:}
1338:s
1335:{
1331:C
1319:C
1295:)
1292:s
1289:+
1284:n
1280:x
1271:2
1267:y
1263:(
1258:}
1255:s
1252:,
1249:y
1246:,
1243:x
1240:{
1236:C
1219:)
1214:n
1210:x
1201:2
1197:y
1193:(
1188:}
1185:y
1182:,
1179:x
1176:{
1172:C
1134:S
1120:S
1099:X
1085:X
1050:S
1040:0
1008:X
996:0
992:X
962:0
958:X
937:0
917:S
897:S
891:X
888::
885:f
863:0
859:X
824:)
819:n
815:x
806:2
802:y
798:(
793:}
788:n
784:z
780:,
774:,
769:1
765:z
761:{
757:C
747:A
724:I
704:}
699:n
695:z
691:,
685:,
680:1
676:z
672:{
668:C
642:I
638:}
633:n
629:z
625:,
619:,
614:1
610:z
606:{
602:C
592:A
522:B
512:S
490:X
478:X
445:B
437:X
408:S
402:X
399::
396:f
362:g
344:(
321:)
316:]
313:2
310:[
302:(
297:0
293:H
279:H
271:g
260:b
252:x
248:y
244:a
241:b
237:b
229:x
225:y
221:h
214:H
210:H
184:)
178:(
173:1
169:H
40:ε
37:P
33:P
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.