Download Algebraic Geometry Sundance 1986: Proceedings of a by Audun Holme, Robert Speiser PDF

By Audun Holme, Robert Speiser

This quantity provides chosen papers as a result of the assembly at Sundance on enumerative algebraic geometry. The papers are unique examine articles and focus on the underlying geometry of the subject.

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Read or Download Algebraic Geometry Sundance 1986: Proceedings of a Conference held at Sundance, Utah, August 12–19, 1986 PDF

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Algebraic Geometry Sundance 1986: Proceedings of a Conference held at Sundance, Utah, August 12–19, 1986

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Additional info for Algebraic Geometry Sundance 1986: Proceedings of a Conference held at Sundance, Utah, August 12–19, 1986

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Proof of Pro]x~ition 2 6 : First, let D : COC' be t h e c o m p l e t e intersection of quadrics. W r i t e ~C for t h e ideal sheaf of C in p r a n d s i m i l a r l y for C' a n d D. Since t h e canonical bundle on D is given b y OaD:OD(r-3), we h a v e b y t h e t h e o r y of liaison t h a t ~C/~D : (~D:~C')/~D : Hom((~C, , OEO = Hom(0c', ~D)(3-r) = ~C~3-r). F u r t h e r , since D is p r o j e c t i v e l y n o r m a l , w e h a v e for e a c h integer k a n e x a c t 0 -* H0 ~D(k) -~ H0 ~c(k) -* H0 w c ~ k + 3 - r ) -* 0 .

8) F 3D ~ (2d- 3)A/2 - BI2 + + 18B - 3CO + co - 11C + - 5A + 4A0,1 T{*A Z~0,1 Indeed, based on the results of [D-H1] w e m a y m a k e the Coniecture. The Picard group of the Severi v a r i e t y W is generated over @ by t h e classes A, B, and C and the classes of the boundary components A 0 and Al,j. 3) above A, B and C are themselves rational 28 linear c o m b i n a t i o n s of CU, TN, TR and the b o u n d a r y c o m p o n e n t s this is e q u i v a l e n t to &0 and Ai, J, Conjecture'. The Picard group of the v ~ r i e t y V of nodal q u r v e s is torsion.

This includes most of the geometric divisor classes studied in this paper. 5) now allows us to c o m p u t e r(A) in two ways. They both come out to be equal to (8 + I ) A . 5). 6) Theorem: Let S(d, 8) c Pic(W(d, 8)) ® Q be the subspace spanned b y A, 2). Then the dimension of S(d, B,C, and A; a s s u m e t h a t 0<_ 8_< ~ ( d - l ) ( d 8) as a vector space over • is: i). dimS(d, O) = i ii). dimS(d, i) = 2 iii). dimS(d, 2) = 3 iv). dimS(d, 8) = 4 for 3 <_ 8 <_ ~(d-1)(d-2) - 2, v). e. g = 1), vi).

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