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

ISBN-10: 3540123377

ISBN-13: 9783540123378

ISBN-10: 3540409718

ISBN-13: 9783540409717

**Read 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**

**Sample text**

Then S s But we have very ample on D, there are curves ~3 . 2 is similar to the above, except that we draw cubics through nine points of H. 0 C. l I wish to thank Joe Harris for suggesting the monodromy arguments used in this paper. §2. Let Ir : X ~ S b y some v a r i e t y nodal if all the Let ~ and suppose S. X If be a flat family of stable curves of genus s ~ S, Xs = 7 r - l ( s ) . We s a y g parametrized ~ : X ~S is equi- have the same number of nodes. s be a line bundle on £s we l e t has degree d.

Suppose +dimS=r(~+l) -(r+1)% and that tion at ha is 6ood. Then ~ o_~n ~ x T 2 ~ ~ satisfies the Brill-Noether condi- a. Proof. 2 shows that every component of dimension at most k(~ + dim S) dim G + kr. passing through r On the other hand, the dimension of T is and the fiber dimension of k(~ + dim S + Ag). ~ T is A kAg. So (r+l)((g+k~g) -(d+kd 0) +r) So the Brill-Noether condition is satisfied. has dim ~ = A short computation shows that the codimension of is at least a Ar in A 58 §4. Let We suppose : C -* p n image of C be a smooth nondegenerate curve in d > r + 2, and let L = @C(1).

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. on X Let is a stable curve of genus S be the set of all g' (X,M) e S S with X singular, S 3 N T = T 3. and let Further, S3 it is easy to see that the moduli of stable curves, which meets T. But if c o d ~ T T 3 ~ p. of degree So one sees S 3 ~ Ss, S 3 ~ S s. d + 2k be the set of all and genus T then Now since g + 4k (X,M) codim S in h0(X,M) = 4.

### Algebraic Geometry: Proceedings of the Third Midwest Algebraic Geometry Conference held at the University of Michigan, Ann Arbor, USA, November 14–15, 1981 by D. Burns (auth.), I. Dolgachev (eds.)

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