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Download Algebraic Geometry Sundance 1986 by Holme R. Speiser (Eds.) PDF

By Holme R. Speiser (Eds.)

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

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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. W(d,S)' 9(d,8) • ~ V'(d,8) ig W(d,8+l) 9(d,8+I) The morphisms nl, n2, n3, and n 4 are normalizations and the morphism g comes from the universal mapping property of normalization applied to n 3 . It is clear t h a t g is proper. W' (d, 8) is obtained from W (d, 8) by adding codimension two subvarieties.

Pic(W(d, 8)) ~ Pic(W(d, 8 + i)). 2) Definitipn: r = g, o n 2 oi x o j . To c o m p u t e t h e h o m o m o r p h i s m r explicitly we need a description of the local s t r u c t u r e of W'(d, 8) n e a r A'(d, 8) . By t h e d e f o r m a t i o n t h e o r y of [D-H2] we see t h a t we m a y obtain this local i n f o r m a t i o n b y looking in the d e f o r m a t i o n spaces of a p p r o p r i a t e singularities. 5) L e m m a . In the d e f o r m a t i o n space of a tacnode y 2 _ y x 2 + tlx2 + t2 x + t3 = 0 the following loci m a y be described as follows.

R e m a r k : We allow t h e possibility t h a t t h e scroll m a y be singular, a n d r e q u i r e of C only t h a t it be a p u r e l y 1-dimensional s u b s c h e m e . i m a y be applied in several situations. I applies. It also applies to all nondegenerate curves of degree ! ,]atainedin a hvperplane, then C is contained in a rational normal scrollQf dimension 2. 5: If D c pr+l is the rational normal curve of degree r+l, and L is a line meeting D in (at least) a point, then C = D U L is contained in a Z-dimensional r a t i o n a l n o r m a l scroll.

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