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By D. Burns (auth.), I. Dolgachev (eds.)

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Read Online or Download Algebraic Geometry: Proceedings of the Third Midwest Algebraic Geometry Conference held at the University of Michigan, Ann Arbor, USA, November 14–15, 1981 PDF

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Additional info for Algebraic Geometry: Proceedings of the Third Midwest Algebraic Geometry Conference held at the University of Michigan, Ann Arbor, USA, November 14–15, 1981

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Hence the 1 points of C O Ci C n C. are determined up to finitely many choices if three points of i coincide with three points of C N R i. Further, if C N C i = C O R i, then C i = Ri, since the conic through five points in this case L and depends on at most short computation Now let and L S M are isomorphic, 4(d - g - 3) + 3k shows that in a general position is unique. since they are both parameters. Let @D(1). Thus Also T3 p = 4(3 + g' - d'). A codim T T 3 ~ p. be the set of all is a line bundle of degree (X~L), where d' X.

W'o. i i Then L 47 is a quadratic extension of a representation of Proof. Clearly given by F(t), where _S4 on t 2 = (w21) (w2)(w3). x4), extension of That M as _S4 on differs V3 k, and if T (yi) = yj, ~4 v wi' wiw i' Yi M ~ k(V~), (i=1,2,3). then M is a k(Sym2V4)~4 = M~3(t,(7), where t = w|w2w 3. k (Sym2V4)~4 = (M(t,(7))~3 , beginning, while 2 be the field generated by purely transcendental Proof. l • (w i) = +wj, hence T(wi) J) Step IV. t = WlW2W 3. ~3 = ~4/G" t # F, from the one of F t is an is isomorphic to ~3 but o is an invariant for follows by step III.

So Thus + dim S = r(fig + i) - (r + l)d 0 . Now for a generic V of dimension satisfies the hypotheses namely (D,@(D)), if of §4, (r + l) in 2k >_ h0(C,L) - (r + i). gives the existence of smooth nondegenerate and genus g + (r+l)k. 1 constructs a good C and of degree g' of genus hi(L)- 2k >_ h0(C,L) 0, d + ( r - l)k satisfying the g and and a line k - (r + 1). 1 is established. 2, we let sional family of elliptic curves and let and d O = 3. The proof of Proposition Vl : C1 -~ S1 Q1 be a nontrivial one dimen- be the zero section.

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