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 examine articles and focus on the underlying geometry of the subject.

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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}. 35 For 8 = 1, TN = 0 so d i m S ( d , i ) s 2. For 8-- 2, T R - - 0 so d i m S ( d , 2)_< 3. S ( d , i ) is s p a n n e d b y CU and NL. S(d, 2) is s p a n n e d b y CU, TN, and NL. The m a t r i x t h a t r e p r e s e n t s r S ( d , I) --, S(d, 2) w i t h respect to these s p a n n i n g sets is: 0 ii:l w h i c h has r a n k 2.

Ince TtC~ is a vector bundle of rank r-1, it follows t h a t r-Z general sections will be everywhere independent on C, and thus the intersection ~ of the corresponding hypersurfaces will have no singularities along C, proving the first statement. To prove the second statement, fix 5 as above. It suffices to show that a general element of V induces a section of the line bundle JIC/S ~ having 60 only simple zeros. But b y o u r hypothesis V g e n e r a t e s JlC/S ~, so again b y a version of Bertini's T h e o r e m a general section v a n i s h e s on a reduced set of points, as desired.

Thus on the base ~1 of t h e f a m i l y we have: deg(A) -- 2 a b - 1 deg(B) = (2g- deg(C) = -I deg(A) = 2)a- i I. Family Three. Consider a generic pencil of plane cubics. All except finitely m a n y of its m e m b e r s will be nonsmgular and t h e rest will h a v e one node. Let X be the blow up of p2 at the 9 base points of the pencil, E l , . . ,E 9 the exceptional divisors and H the pullback of the h y p e r p l a n e class on p2. X is a f a m i l y of elliptic curves. Let i be a subset of {1, 2 .

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