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

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

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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

Example text

Xmn ~, and a perm Vn mutation ~ acts on ant rational xij functions, and a variable by sending it to xi~(j ). where o. 2. m. 1 form a basis is a rational of the purely field: more precisely transcendental extension k. o', 0 (2) . . . Proof. ~O The invariant functions invari- function, h = 2 .... m o(h) n = E "'''Xlj' i j=l Xhj ~i-l(Xll . . n (h) n o I = Ni I XhjXlj (2. vm n ÷ (An)m a morphism onto the affine Vm n T e S such that ~(x) = y. ,n). 1, then, in particular, The system being given by the matrix I Z1~ ...

Let V' is be are contained in a smooth cubic curve. Pr+3 ~ ~V '(C) in H V ~ ~(C,L) ~V : C -~ ~ r is ~eneric linear subspace dimen- h0(L) > 10, be a subspace of and If we let H. has only nodes and degree R q + 1 = h0(C,L). and if but nine points of all lie on a hyperplane V and be a very ample line bundle of dim V = r + l, is a ~eneric hyperplane in r = 3 L W = ~(C,L), h0(C,L) >_ r + 4 If in a cubic c u r v % Proof. we have h0(O,%(1)) :4. 1. 1. Let d so that H ~ C generically and choose elliptic curves are chosen 6enerically, a linear system without base points with sion It then any nine points of ~(E i n C) : 9.

We can assume from now on Let ]p2, < (3H-4) 2 - (3H-£)'F. - rj e 0 since i > j). M 2 -> (3H-A) 2 = 9 - line in i > j. M 2 = (F+M) 2 - ( F + M ) ~ F - F M Hence If F is a sum of exceptional Ei, ] given by the linear system in only one point, has no vanishing Let Now ~:S ÷ ~ 3 ÷ H°(0H,(3H-4)) as a twisted cubic. of which intersect F the singular points the exact sequence ÷ H°(0s(3H-4)) says that the rational map H' through Q n G, let IH - L I one has a restriction q on be the induced divisor: G .

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