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

ISBN-10: 3540123377

ISBN-13: 9783540123378

**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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**Extra 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**

Let us choose affine 4 map of which IP3 . leaves two involutions of X {(~,o~), points. if we take the linear automorphism ]pl c M4, 1 p = (~,~), in three distinct nant map of divisor set of five points Given a general ~,1 rational that a curve of genus 4 r g3 = g|g2' is the quotient of g3(x,y) = (l-x,y), 4; with ~10 then g = (y,x), g2(x,y) if we set and our group our discussion ]pl × ]pl _ A: such that gl(x,y) has period on belongs and by the = (x,l-y). s = g3' is the dihedral then group s 2 = I, D 4.

I=l i=l - then the only other possibili- = 0X(n) , in = O, we conclude that 0. [4H-241: since D; ÷ 0 since D m 2(2H-A), hence so that G G D o X = ~, D 12H - A[ = 9, is mapped S admits a double admits a double covering is a symmetric we have proven that G to the ramified at cubic. induces on X the double cover ~. 15) D in the form r 2 = 2. 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.

Theorem hi(F ) the 32 §I. GEOMETRY OF CURVES OF GENUS Let [KxI X be a non-hyperelliptic gives an embedding of intersection of a quadric The quadric Q X Q curve of genus in ~3 4. 4. Then the linear system such that the image of and of a cubic X is the complete G. is uniquely determined, and, since it is normal, as well as G, there are only two possibilities: i) Q is smooth ii) Q is a quadric cone. e. 2~ m KX) such that h (0X(~)) = 2: we say then that X has a vanishing thetanull. I. Let form an n ~ Pic2(X) - {0}.