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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Additional info for Algebraic Geometry Sundance 1986: Proceedings of a Conference held at Sundance, Utah, August 12–19, 1986
Q 2 - deE(A) = 2 a b - t l I deE(B) = 4b- 2 a - III ~ E. i I= 1 40 deg(C) = 28 deg(A) = 20. The proof of independence n o w follows i m m e d i a t e l y f r o m the existence of these families. Specifically, to show a) a b o v e use families i and 2; for b) use families 1, 2 and 3; for c) use families 1, 2, 4 and 5 a n d for d) use families i a n d 4. ~ I e s In this section, we will consider a series of special cases, where the results of [D-HI] and of this paper either yield results about families of plane curves, or m a y be verified directly.
T 1 = - 2 a , 3. 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.
F) = 6d-6, with a point of multiplicity (F. Ei) -- 3 a t each of the base points of t h e pencil Z. These, together w i t h t h e nodes of ~ at the singular points of the c u r v e s in Z are all the singularities of ~. 45 Indeed, we can c o m p u t e the geometric genus of can apply the genus formula to ~ c [p2 to find g(~) = (6d-7)(6d-8)/2 in two ways. d 2 - 3(d-l) 2 = 12d 2 - 39d + 25. Alternately, we can realize the n o r m a l i z a t i o n of ~ as a b r a n c h e d cover of Z p l with 3d(d-2) sheets, b r a n c h e d simply over the divisor HF and h a v i n g 4 b r a n c h points (two points of ramification index 3) over each point of A (cf.
Algebraic Geometry Sundance 1986: Proceedings of a Conference held at Sundance, Utah, August 12–19, 1986 by Audun Holme, Robert Speiser