# Basic Algebraic Geometry 1 - Vars. in Projective Space by I. Shafarevich By I. Shafarevich

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Moreover, at the singular = 3. Applying above gives a symmetrization. Conversely, consider of Z', f:Z ÷ G and define Y f~ Y --+ G let of type An, be a n = 1,3,5. desingularization is a Cohen-Macaulay a reduced p:G ÷ G, of G, section of sheaf on  of with algebra we prove 0 + -03]p3 effective G in and G. D. now that if we write L 0~(e) cor. 2. 17. 19 of the sheaf Now, G. It follows h°(0~(H-L)) to be is a minimal has only nodes where therefore (cf. , prop. and list. has only singularities where in the exceptional 2L m E, ~Z = f*(-H) + f*(L), G the plane line so that this cubic has only one symmetrization.

E. 2n m0, n ~ 0. e. is the linear system of effective group in n letters, 0y(Ky) ~ ~y. divisors D' m D. R. U is a HI(Y,F) F as a representation k-vector k n. Y, we denote by space. k-vector space, we denote by is an abbreviation on on a complete variety U* for the Riemann-Roch its dual space. 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.

Now let 0 A r = [a c 3 I h (~p(a),~a) = r + i] . We obtain a map ia : HO(2p(a),~a) ~ HO(c,L) , determined up to multiplication by a non-zero element of tive. Indeed, any section of nodes of ~p(a)' the Grassmannian a well-defined ~a which vanishes on and hence must vanish on of map algebraic varieties. ,Csk subspaces of k. We note ia is injecmust vanish on all the since ~ > dO . H0(C,L). We leave the reader to check there is a morphism ¢2(a) = (Pll(t) ..... Plr(t), ¢1 ¢ 2 : A r ~ C kr. Let G be We have obtained is a morphism of Indeed, if p(a) = t, P12(t) .....