By Audun Holme, Robert Speiser
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 subject.
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Extra resources for Algebraic Geometry Sundance 1986: Proceedings of a Conference held at Sundance, Utah, August 12–19, 1986
Example 2: reducible c u r v e s We will exhibit here some of the techniques for dealing with families of reducible c u r v e s b y considering the f a m i l y of c u r v e s formed b y taking a general c u r v e C c p2 of degree d-1 and genus g h a v i n g $ = ( d - 2 ) ( d - 5 ) / 2 - g nodes and adding a variable line Lx m o v i n g in a pencil. 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.
H0~C ' @ HOOc~k) -* H0~c~k) ) are surjective for all k>O. 7: If C is a reduced and irreducible locally Gorenstein curve of arithmetic genus >0, and L is a line bundle generated by its global sections with hO(L)_~5, then the module ~HO(y_ -" L n) is generated, over the ring ~H OLn bV elements of degree i O. 4: Bertmi's Theorem s h o w s t h a t neither ~ nor D have singularities a w a y from C. Let 3lC~ = JC/~C2 be the conormal bundle of C, and let V = H0~C(2). Our hypothesis implies t h a t V generates ~C~(2) on C.
T 3 = 0, t4 = 0 2. t 2 = 0, t4 = 0 3. t 2 = t3, t I t 2 = t 4. o b s e r v e t h a t w h e n w e pull t h e s e loci b a c k to t h e ( r , s ) - p l a n e , t h e l o c u s of c u r v e s w i t h t h r e e n o d e s is g i v e n in b r a n c h 1) b y r = 0, t h e locus of c u r v e s w i t h a t a c n o d e b y s 2 = 4r; and that these have intersection multiplicity b r a n c h 2) t h e s e t w o loci a r e g i v e n b y t h e e q u a t i o n s respectively, and have intersection number s2 = - 4 r , s2 = 4r 2; a n d in b r a n c h 3) b y r -- 0 a n d again having intersection multiplicity m u l t i p l i c i t y of t h e s e t w o loci is t h u s r = 0 and 2; s i m i l a r l y in 2.
Algebraic Geometry Sundance 1986: Proceedings of a Conference held at Sundance, Utah, August 12–19, 1986 by Audun Holme, Robert Speiser