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Double solids, Advances in Mathematics, to appear. Publ. S. 10 (1929) (reprint 1961). : The unirationality of A 5, to appear. Harris, J. 67, I(1982) 23-86. : Arithmetic variety of moduli for genus two, Annals of Math. 72(1960), 612-649. : The uniruledness of MII , to appear. : Geometric invariant theory, Springer Verlag (I~65). E. Varenna, 1969, Cremonese. : Prym varieties I, in "Contributions to Analysis", Academic Press (1974), 325-350. : La variedad de los modulos de curvas de genero 4 es unirracional, Ann.

We have already proven that giving G symmetric containing X determines n ~ Pic2(X) - {0}. 6) IK + HI Proof. ~:X ÷ ~2 given by the linear system We break up the proof in several steps. Let has no base points if p ~ X: if H1(0x(K+n-p)) if q e IP-~I, since = 0. (0X(K+n)) X = 0 By Roch's duality, 2p m 2q, with is not hyperelliptic. p is not a base point this is equivalent q # p, and X to if and only IP - nl # ~. But is hyperelliptic. D. 7) Proof. ~ is a morphism. degC ~ 3 If C Denote by if X C = ~(X), h°(0x(D)) and then, let e 2, D' ~ D, and since D D be the inverse image of a general has degree 2D m K + ~ , base points, by the "base point free pencil trick" , H o (0x(D)) ~ H o (0x(D)).

Y4) ramification point of p, C A in the points cor- p, we define analogously F' c C 2. and and the general. then ~ W Yl = Y2 and it follows that Yl is a ramification point Y3 = Y4' hence Y3 is a second p'(yl ) = p'(y3). It is easy to see that curves C of type tion can hold form a proper subvariety ' n F23 ' FI2 n F34 n FI4 To show that to F~k = (fh×fk)*(F'). 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.

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