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

By Audun Holme, Robert Speiser

This quantity offers chosen papers due to 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

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T l = - ~ d , t 2 = 2d 2, t5 = t4 = 0 t 3 = - a 2, t2 = t4 = 0 t2=t3=-ld 2. t4= and 3 or in Cartesian f o r m b y equations: 1. t 3 = 0, t 4 = 0, t 2 = 2t 2 2. t 2 = 0, t 4 = 0, t 2 = 4t 3 3. t 2 = t3, t I t 2 = t4, t 2 = - 4 t 5. The locus of c u r v e s w i t h two nodes likewise has t h r e e branches, given either p a r a m e t r i c a l l y as 1. t I = s, t 2 = r, t 5 = 0, t4 = 0 2. t 1 = s, t 2 = 0, t 3 = r, t4 = 0 as 32 3. t 1 = s, t 2 = r, t~ = r, t4 = r s or in C a r t e s i a n f o r m b y e q u a t i o n s : 1.

W i t h $+d-1 nodes) parametrized by X ¢ p1 To begin with, the s i m u l t a n e o u s n o r m a l i z a t i o n C will consist of t w o disjoint components, the product X 1 ~- C × p1 of the normalization C of C w i t h the parameter c u r v e p 2 and the ruled surface X2 ~ ~-1 swept out b y the lines Lt. On X 1 both the divisor classes D and co are pullbacks f r o m C, and so all pairwise products a r e zero. On X2 the Picard group is generated b y the class of a line L = Lx (that is, a fiber over p1) and the exceptional divisor E, with the divisor D equivalent to L*E and the class of the relative dualizing sheaf co - -L-2E.

B A. References [D] S. S. 56 (1985) [D-HI] S. Diaz and J. Harris, Geometry of the Seven variety, preprint [D-H2] S. Diaz and J. Harris, Ideals associated to deformations of singular plane curves, preprint IF] W. eme-theoretic versus homogeneous generation of ideals Lawrence FAn, David Elsenbud, and b-~heldon Katz* Conten~ Positive results 1) C u r ~ on rational normal scrolls 2) Curves m Pq and p5 (Counter-) Examples 3) Deccerminantal constructions 't) General sets of points 5) Elliptic octic curves in p5 Summary In this note we consider cases in which a c u r v e in p r which is scheme theoretically the intersection of quadrics necessarily has homogeneous ideal generated by quadrics.

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