36:
186:, because of the possibility of multiple components. For (irreducible) varieties, if one takes into account the multiplicities and, in the affine case, the points at infinity, the hypothesis of
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may be replaced by the much weaker condition that the intersection of the variety has the dimension zero (that is, consists of a finite number of points). This is a generalization of
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The degree is not an intrinsic property of the variety, as it depends on a specific embedding of the variety in an affine or projective space.
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In the affine case, the general-position hypothesis implies that there is no intersection point at infinity.
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Hilbert series and
Hilbert polynomial § Degree of a projective variety and Bézout's theorem
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The degree can be used to generalize BĂ©zout's theorem in an expected way to intersections of
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228:, then the degree of the intersection is the product of the degrees of the hypersurfaces.
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243:. It follows that, given the equations of the variety, the degree may be computed from a
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has repeated factors, that intersection theory is used to count intersections with
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is an extrinsic quantity, and not intrinsic as a property of
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is the number of intersection points of the variety with
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The degree of a projective variety is the evaluation at
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defining the embedding by its space of sections. The
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182:, the intersection points must be counted with their
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212:of its defining equation. A generalization of
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389:will be equal to that dimension. The degree
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64:. Unsourced material may be challenged and
222:projective hypersurfaces has codimension
490:. The degree can also be computed in the
128:Learn how and when to remove this message
458:For a more sophisticated approach, the
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216:asserts that, if an intersection of
62:adding citations to reliable sources
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486:. The degree determines the first
359:{\displaystyle \dim(V)+\dim(L)=n.}
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27:Number used in algebraic geometry
510:an appropriate number of times.
77:"Degree of an algebraic variety"
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438:defining it (granted, in case
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401:has an (essentially unique)
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18:Degree (algebraic geometry)
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514:Extending BĂ©zout's theorem
506:intersecting the class of
462:defining the embedding of
272:algebraically closed field
460:linear system of divisors
184:intersection multiplicity
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476:tautological line bundle
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427:= 0 is the same as the
502:, with the class of a
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270:and defined over some
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403:embedding of degree
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251:of these equations.
194:. (For a proof, see
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559:Algebraic varieties
397:. For example, the
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154:projective variety
522:hypersurfaces in
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16:(Redirected from
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492:cohomology ring
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56:Please help
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488:Chern class
468:line bundle
383:codimension
172:hyperplanes
142:mathematics
504:hyperplane
416:Properties
381:, and the
255:Definition
88:newspapers
500:Chow ring
375:dimension
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369:Here dim(
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178:. For an
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45:does not
553:Category
446:, as in
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51:sources
150:affine
148:of an
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530:Notes
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249:ideal
109:JSTOR
95:books
259:For
81:news
49:any
47:cite
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478:on
470:or
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408:in
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333:dim
315:dim
300:in
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174:in
156:of
152:or
140:In
60:by
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526:.
412:.
524:P
520:n
508:V
496:P
484:V
480:P
464:V
440:F
436:F
425:F
410:P
405:n
395:V
391:d
387:L
379:V
371:V
354:.
351:n
348:=
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342:L
339:(
330:+
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324:V
321:(
298:L
291:K
287:V
283:V
279:d
275:K
268:P
261:V
233:1
225:n
219:n
168:n
162:n
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125:(
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116:(
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