By Audun Holme, Robert Speiser

This quantity provides chosen papers due to the assembly at Sundance on enumerative algebraic geometry. The papers are unique learn articles and focus on the underlying geometry of the subject.

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

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Introduction Several i m p o r t a n t results in the t h e o r y of projective curves assert t h a t a given class of curves has homogeneous ideal generated by quadrics. Such for example is the case of a canonically embedded c u r v e (Noether's Theorem) or a curve e m b e d d e d by complete linear series of high degree c o m p a r e d to the genus of 52 t h e c u r v e . Because direct g e o m e t r i c t e c h n i q u e s a r e available, t h e s e results a r e generally easier to p r o v e s c h e m e theoretically - - in algebraic language, it is easier to p r o v e t h e w e a k e r s t a t e m e n t t h a t t h e ideal g e n e r a t e d b y t h e q u a d r a t i c a n d linear f o r m s v a n i s h i n g on t h e c u r v e agrees w i t h t h e ideal of t h e c u r v e u p to a n "irrelevant" c o m p o n e n t .

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). e. g = 0). Proof: W(d, 0) is IPN with a set of codimension 2 r e m o v e d so clearly d i m S ( d , 0) = i. 4) we h a v e span(A, B, C, A} = span{CU, TN, TR, A} = span{CU, TN, TR, NL}.

E 9 . We m u s t s u b t r a c t f r o m this the pullback of the canonical bundle of t h e base of the family, which is m i n u s two fibers; we h a v e co = - 3 H + E i + ... E 9 - ( - 2 ( 3 H - EX - ... -Eg. Thus on the base of the f a m i l y we h a v e deg(A) : m 2-1II deg(B) = 3 m - I I I deg(C) = O. To c o m p u t e the degree of A in this f a m i l y recall the following w e l l - k n o w n l e m m a (see for e x a m p l e Diaz [D]). 7) L e m m a : Let r~ : S -* C be a flat f a m i l y of c u r v e s w i t h S a nonsingular surface, C a nonsingular c u r v e a n d all fibers either nonsingular or w i t h o r d i n a r y nodes as t h e i r only singularities.

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