## Download PDF by D. Burns (auth.), I. Dolgachev (eds.): Algebraic Geometry: Proceedings of the Third Midwest

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

ISBN-10: 3540123377

ISBN-13: 9783540123378

ISBN-10: 3540409718

ISBN-13: 9783540409717

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

**Example text**

T = [h e M r I C h Namely, satisfies be the }filbert scheme of all curves We also denote the dualizing sheaf of ¢*(~%) S 7r : X ~ S. ]pn be the tautological line bundle on we obtain a morphism Then S. and genus n = d + 3(2g - 2) + 1 - g. s. denote the0 open set of be the inverse image of Mn. s in M ~. s. is smooth and let is dense in ID. 54 Indeed, let w0 e ~ and let C0 and Assume first that the support o f DO DO be the corresponding curve and divisor. is disjoint from the singular locus of We can find flat family of curves v : C ~S' and a relative Cartier divisor on Cs0 = C O and Ds0 = D a neighborhood of shows that determines & m a p Suppose DO V : S' ~ M V' : S ~ ~ CO.

In fact, if since is the full symmetric group, transversal W - AI2 - A34 - Al4 - A23 = FI2 n F34 n F'14 n F'23, is transversal, eral, we consider the variety (3,3) in Q such that the above situa- in the linear system gives a transversal A c [0Q(3,3)] A = {(C, yl,Y2,Y3,Y4 ) I Yi ~ C, A has only ordinary p. e. 4, × Q4 i0Q(3,3)] intersection for C gen- defined by p(yl) = p(y2) , p(y3) = p(y4) , p'(y2 ) = p'(y3) , p'(yl ) = p'(y4)} is of dimension 15 and smooth at the general point, hence our assertion is proven if the projection of the case, for C A general, Finally we compute: on [0Q(3,3)] is surjective: FI2 n F34 n Frl4 n F'23 but if this were not would be empty.

U = x2/ 2 = x4/ 2 2 . 1 Then but is not linear any more: eigenvalues equal to The quotient is obviously rational. D. (+I), so that Zg/2 4 on an equal to s(w) = -w. 8-dimension (-I). 50 REFERENCES [lJ [2j [3j [4j L5] [6] [7J [8j [9j [10J Lll] [12] [13j [14j [15J ~16] [17j [18j Arbarello, E. : The equation of a plane curve, Duke Math. J. 46, 2(1979), 469-485. Math. 63(1981), 433-465. Catanese, F. Alg. 23(1982), 1-12. : Double solids, Advances in Mathematics, to appear. Publ. S. 10 (1929) (reprint 1961).

### 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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