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

By Audun Holme, Robert Speiser

This quantity provides chosen papers caused by the assembly at Sundance on enumerative algebraic geometry. The papers are unique study articles and focus on the underlying geometry of the subject.

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

Example text

2) has degree 8 + 1. The o t h e r four equalities a r e easily seen to be set theoretically true. W h a t r e m a i n s is to v e r i f y the multiplicities. Let C be a reduced irreducible c u r v e whose singularities a r e e i t h e r 8+1 nodes, 8 nodes and one cusp, 8-1 nodes and one tacnode, or 8-2 nodes a n d one triple point. Let q be t h e point in FN corresponding to C. Label the singular points of C PI . . . Pro. Let B i be the base of the etale v e r s a t d e f o r m a t i o n space for t h e s i n g u l a r i t y of C a t Pi.

4) of § i to express A, B, C, and A as linear combinations of CU, TN, TR and NL. 5) can be used to c o m p u t e the image u n d e r r of a n y class in the span of A, B, C and A. 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).

2 R e s t r i c t i o n maps a n d i n d e p e n d e n c e of d i v i s o r c l a s s e s In this section we define for each d and 8 a homomorphism r : Pic(W(d, 8)) -~ Pic(W(d, 8 + 1)). We also compute r explicitly on the span of the classes A, B, C, and A. This allows us to determine when the classes A, B, C, and A are independent. Recall the definition of V(d, 8) in ~1. Let V'(d, 8) be V(d, 8) U V(d, 8+1) Define W'(d, 8) to be the normalization of V'(d, 8) and A'(d, 8) to be the inverse image of ~/(d, 8+1) in W'(d, 8) with its reduced scheme structure.

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