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

By Holme R. Speiser (Eds.)

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 topic.

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E. 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.

5: The first r - 1 of t h e h y p e r s u r f a c e s c u t t i n g o u t C m e e t in CUC °, so t h e r t h m u s t m e e t C' in e x a c t l y t h e (r-3)d - 2g + 2 points of intersection. Thus this n u m b e r m u s t be twice t h e degree of C~. The f o r m u l a s of Proposition 2 2 a n d simple a r i t h m e t i c n o w yield t h e de~sir~l r ~ u l t . Proof of Pro]x~ition 2 6 : First, let D : COC' be t h e c o m p l e t e intersection of quadrics. W r i t e ~C for t h e ideal sheaf of C in p r a n d s i m i l a r l y for C' a n d D.

3: If Cc p5 is a proj~tiv~Iy normal c u r w which is the ~ h e m v theoretic intersection of quadrics, then the homogeneous ideal of C is generated by quadrics. To p r o v e these results we will use t h e residual c u r v e C' to C in t h e complete intersection of r-1 general quadrics containing C. The following result shows us (in a m o r e general setting) w h a t we c a n expect of C'. Its first s t a t e m e n t will b e c o m e r e l e v a n t in section 5. 3. 4: If C c p r is a s m o o t h c u r v e w h i c h is t h e s c h e m e t h e o r e t i c intersection of h v o e r s u r f a c e s of degree 2, t h e n a ~eneral set of r-Z a u a d r i c s containing C m e e t in a s m o o t h surface, a n d a general set of r-1 such quadrics m e e t in a c u r v e of t h e f o r m CUe', w h e r e C' is a s m o o t h ( t x ~ i b l y e m p t y or disconnected) c u r v e m e e t i n g C in o r d i n a r y double points.

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