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More specifically there are two closely related types of quartic surface: affine and projective. An
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608:(The union of a cubic surface and a plane is another particular type of quartic surface)
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602:(The union of two quadric surfaces is a special case of a quartic surface)
529:. If on the other hand the base field is finite, then it is said to be an
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respectively. One must be careful to distinguish between algebraic
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On the other hand, a projective quartic surface is a surface in
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quartic surface is the solution set of an equation of the form
359:{\displaystyle f(x,y,z,w)=x^{4}+y^{4}+xyzw+z^{2}w^{2}-w^{4}}
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199:{\displaystyle f(x,y,z)=x^{4}+y^{4}+xyz+z^{2}-1}
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16:Surface described by a 4th-degree polynomial
240:of 4 variables of degree 4, so for example
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503:surface given as a quartic curve over
657:Quartic surfaces with singular points
106:is a polynomial of degree 4, such as
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628:, Cambridge Mathematical Library,
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574:are examples of quartic surfaces.
567:=0 (an example of a K3 surface).
660:, Cornell University Library,
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520:{\displaystyle \mathbb {C} }
486:{\displaystyle \mathbb {R} }
469:, and quartic surfaces over
460:{\displaystyle \mathbb {C} }
414:{\displaystyle \mathbb {C} }
388:{\displaystyle \mathbb {R} }
92:{\displaystyle f(x,y,z)=0\ }
705:
630:Cambridge University Press
531:arithmetic quartic surface
423:the surface is said to be
229:of the same form, but now
35:defined by an equation of
625:Kummer's quartic surface
570:More generally, certain
537:Special quartic surfaces
208:. This is a surface in
654:Jessop, C. M. (1916),
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495:. For instance, the
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620:Hudson, R. W. H. T.
689:Algebraic surfaces
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25:algebraic geometry
667:978-1-4297-0393-2
639:978-0-521-39790-2
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684:Complex surfaces
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29:quartic surface
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578:Kummer surface
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549:Fermat quartic
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543:Dupin cyclides
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497:Klein quartic
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210:affine space
101:
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28:
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572:K3 surfaces
551:, given by
236:homogeneous
21:mathematics
678:Categories
613:References
238:polynomial
344:−
191:−
622:(1990),
594:See also
648:1097176
527:
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432:complex
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33:surface
664:
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102:where
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44:affine
37:degree
499:is a
443:over
233:is a
31:is a
662:ISBN
634:ISBN
547:The
501:real
426:real
27:, a
429:or
397:or
39:4.
19:In
680::
644:MR
642:,
632:,
563:+
559:+
555:+
533:.
368:.
217:.
565:w
561:z
557:y
553:x
514:C
480:R
454:C
408:C
382:R
352:4
348:w
339:2
335:w
329:2
325:z
321:+
318:w
315:z
312:y
309:x
306:+
301:4
297:y
293:+
288:4
284:x
280:=
277:)
274:w
271:,
268:z
265:,
262:y
259:,
256:x
253:(
250:f
231:f
226:P
214:A
194:1
186:2
182:z
178:+
175:z
172:y
169:x
166:+
161:4
157:y
153:+
148:4
144:x
140:=
137:)
134:z
131:,
128:y
125:,
122:x
119:(
116:f
104:f
84:0
81:=
78:)
75:z
72:,
69:y
66:,
63:x
60:(
57:f
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