By D. Burns (auth.), I. Dolgachev (eds.)
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 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
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] .
Sion If r - l, Let Hi H'. - U R~ or c o d ~ v , V" = 2. and let V~ = H0(D~,@(1)). are generic, then either the codimension of Indeed, let V' = V~ and V~+ 1 V" = ~(D~+I,@(1)). r - 2 passes through these points. V~ is two or Note that the H"s ~ ~V ''(C) be in a linear subspace of dimension degree in We claim that if the r - l, Hence (r + 2) V~+ 1 = V. points of since a rational curve of V" = V or cod~, V" = 2. 1 is established. Now consider the case in H 0 C r = 3- We suppose that there is an and a nonsingul~r cubic curve in further assume that not all the points of H H N C E.
Tion M4, 1 is a rational variety. 19 and the arguments preceding it we have a linear representa0:D 4 ÷ Aut(U) where U is l]-dimensional, nal to ~(U)/D4. 6) s(x,y) = (y,x), To decompose assume For s For sr, U r, s M4, ] isbiratio- 2 2 2 2 2 2 I, x, x , y, y , xy, x y, xy , x y , are the generators of D4, such that r(x,y) = (y,l-x). as a direct sum of irreducibles, char(k) # 2, and we know that we compute the character X since of D4 has order 8 and we O. we observe that O(s) permutes the elements of the basis, leaving 22 I, xy, X y fixed: hence X(S) = 3.
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.)