By Holme R. Speiser (Eds.)

This quantity offers chosen papers caused by the assembly at Sundance on enumerative algebraic geometry. The papers are unique learn articles and focus on the underlying geometry of the topic.

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0 31 N o t e t h a t t h e i n v e r s e i m a g e in t h e ( s , c ) - p l a n e of t h e t w o node locus is g i v e n = ~ c 2, while the cuspidal locus is given b y s = h a v e intersection multiplicity 2. 4) L e m m a . by s c2; and t h a t these two c u r v e s In t h e deformation space of a triple point x 2 y + x y 2 + tlxY+ t2x+ t3Y+ t4 = 0 t h e f o l l o w i n g loci m a y be described as follows: t h e locus of c u r v e s w i t h t h r e e nodes m a y be given p a r a m e t r i c a l l y b y t I = c, t 2 = t 3-- t 4 = 0 or in Cartesian f o r m by the equations t 2 = 0, t 3 = 0, t 4 = 0; t h e locus of c u r v e s w i t h a t a c n o d e has t h r e e b r a n c h e s , g i v e n p a r a m e t r i c a l l y 1.

3) deg(TR) = (d 2 - 6 d + 8 ) / 2 = (d 2 - 5d+6)/2 (g-l) + 1 + (d-6)/2 - g, w h i c h is the n u m b e r of nodes on C. 5) the n u m b e r of flecnodes occurring in the f a m i l y (Cx} is deg(FN) = 6 d + 6 ( g - ! ) - 2 1 - ( 3 d - 18) = 6(d-l) + 3(2g-2), w h i c h is the n u m b e r of flexes of C. Example 3: The case 8 = 0. Of course, as has been observed, in case $ = 0 the Severi v a r i e t y W is j u s t open subset of pN consisting of smooth c u r v e s and c u r v e s with one node; the Picard group of W is generated b y the class A = Cl((gpN(1)).

This allows us to r e d u c e our independence s t a t e m e n t to the case of small v a l u e s of g, w h e r e we. m a y v e r i f y it b y exhibiting c u r v e s in the Severi varieties and explicitly c o m p u t i n g their intersection n u m b e r s w i t h the divisor classes A, B, C and A. Then, in t h e t h i r d section, we will consider s o m e special e x a m p l e s of t h e relations above, as applied to v a r i o u s o n e - p a r a m e t e r families of plane curves. F i n a l l y , w e m a k e t w o observations.