Algebraic Geometry and Complex Analysis: Proceedings of the by George R. Kempf (auth.), Enrique Ramírez de Arellano (eds.)

By George R. Kempf (auth.), Enrique Ramírez de Arellano (eds.)

From the contents:G.R. Kempf: The addition theorem for summary Theta functions.- L. Brambila: lifestyles of definite common extensions.- A. Del Centina, S. Recillas: On a estate of the Kummer type and a relation among moduli areas of curves.- C. Gomez-Mont: On closed leaves of holomorphic foliations by way of curves (38 pp.).- G.R. Kempf: Fay's trisecant formula.- D. Mond, R. Pelikaan: becoming beliefs and a number of issues of analytic mappings (55 pp.).- F.O. Schreyer: yes Weierstrass issues occurr at so much as soon as on a curve.- R. Smith, H. Tapia-Recillas: The Gauss map on subvarieties of Jacobians of curves with gd2's.

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T . Denote by m = w + + ~the u s u a l deI 3 c o m p o s i t i o n of H°(~,~) w i t h r e s p e c t to i. + + _ We h a v e t h a t ~ii = ~ii ' w22 = ~22 ' ~12 = ~12 and ~12 = ~21 " This f o l l o w s from the fact that (~ii) = ~*(D + D'), (~22) = ~ * ( E + E ' ) , (~12) is of = ~*(D + E') and (~21) = ~*(D' ~Ii _ det ~21 + E). e. + + ~ii~22 - ~21~12 and f r o m h e r e ii) Let D, E E C it f o l l o w s i E Y, (4) are d i s j o i n t Let i ~ ~ that ® i-I. e. let be the a b e l i a n d i f f e r e n t i a l and (s~) = D', is a l w a y s by (el2) [-i = ~4(E) = E' Let t) (s 2) = E As b e f o r e si ® i ~j = D + E' = which the case).

A k + l Y k + -(ak+iS) ~- 0 (~/~X i , ~/~Yj) ~t 0 Theorem , , (x0 = (0 , 0 , y l , . . e. 2: x l ' etyl is I) w h i c h Yk+l ) the d i a g r a m in our c h o s e n bi-weighted L(l,0)+L(-al,l)+ basis Euler is 6. sequence: ..... (E)÷0 57 ~_~I {0} ~2-{0} x 1,(El ,~ (x0 , x~ , y~ ..... Yk+l ) -~ ( ~[0 Uoxa:~*,-~UoX/A~l ) a x I 2y 2 ak+ 1 x i Yk+l . . . 1 Xl 0 .... ' ~YI " ~P ) ~Yk+l Xo X21 0 0 . a2 - a a2 - a I _Xl iy2 XI ....... (Yl) 2 Yl Y2 a2 - al - 1 ~-- (a2 - a I)X i . 0 . 0 0 Yk+1 _ Xak+l - a I - 1 YI - (ak+l al) i ~ + I -el - XI Yk+l (y1) 2 S -1 ~l~X0 .

1)Chap. 2~0)of[ACGH3 . smooth, divisor of irreducible J(R), curve of then for a reg @)reg) = Sing 0P(TCx(@))) ]P (To(J(R)) ) . 5). 6. (1) ---OE(U) Proof. From the diagram and the fact that ~ relation between Gauss if w e ) j(~) C ' J (C) is a l o c a l o ~ that , G = s o a' G: x" r~l* e'* seen in is i n j e c t i v e In o r d e r the morphism [acl* and map. 3) that a' ¢'*0E(U) ramified) the results of Beauville, i(~) ~ ~. 1. ® ~' the we need and so claim. 7. n*0H(1) ~ OE(U ® n). 8. Proof. ~'*(n*(0 the above Frorosition will follow from (i)) -~ ~X' Let us first observe volution, then j(g~) m': X' ~ IU G nl* ~ 2 that is m*(0 ~2(i)) = KX, - g~ .

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