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Extra 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
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~ ... Zn ) ,Ol(Z 1 .... ,Zn_ l) On_l(Z 2 .... z n) ,°n_l(Z I .... n. Then the variables of the same system of they are equal system is non-zero. 3) = ~(y): = ~'(y). 3) n is not identically zero. We prove this by induction on det(Z 1 \z2 z21 2 2 Zl" = z I - z 2.
Isomorphism then ~ is in- If H" of H°(0%(H-L))~ ÷ H°(0H,(H-L)), is a general hyperplane sec- 38 therefore, if we denote still by embeds of G map H" ~ as a smooth plane cubic. there passes a plane section 4 : ~ 2 -> G clearly the rational map is given IK + n[ by ~:~ ÷ ~ 2 given by IH - L I, Since through any two general points H" as above, ~ is birational, a system of plane cubics. 9), is not bi- elliptic. D. Just for completeness, we indicate, for the three types of symmetric which are the systems of plane cubics giving the rational map In case ii) we consider ~2 , the six points of intersection of four independent lines in 6 S ~ G, with A = i~iEi, and D £ 14H- 2A I given and we blow then up to get by the of the proper transforms In case iii): ~2 2 of cubics, 4.
3 Thus i #(R i n C) = r + 2. Now let H i, D be any curve of degree R i n D = ~. _< r in ]pr. We claim that for generic Indeed let F = [(Pl ..... c, through We claim r ~+2" through all the Suppose not. Pj except Pi" Now and that H n D is finite. Hence meets H D each R. l Let has degree V' ~_ V a basis for V' plane in ]pk we choose s D in at least Consequently~ for generic Hi, Pick Then Pl ..... ,Pr+3 e H N C. D i N Dj ~_ [Pj]. meets each r + 2 points. R i n D = ~. Di Let meets Di We may assume and D] .