# Introduction to Algebraic Geometry by Dolgachev By Dolgachev

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1. Each point xi lies on four lines which contain two other points xj 6= xi . 1) lies on the line T0 = 0 which contains the points (0; 1; ); (0; 1; 2 ) and on the three lines T0 ? T1 ? T2 = 0 which contains the points (1; 0; ); (1; ; 0). The set x1 ; : : : ; x8 is the needed con guration. One easily checks that the nine points x1 ; : : : ; x9 are the in ection points of the cubic curve C (by Remark 1 we expect exactly 9 in ection points). The con guration of the 12 lines as above is called the Hesse con guration of lines.

W Pm (K ) of quasi-projective algebraic ksets is called regular if there exists a nite open cover V = i Ui such that the restriction of f to each open subset Ui is given by a formula: x ! (F0(i)(x); : : : ; Fm(i) (x)); where F0(i)(T ); : : : ; Fm(i) (T ) are homogeneous polynomials of some degree di with coe cients in k. Proposition 2. If V = X (K ) and W = Y (K ) for some projective algebraic k-varieties X and Y , and f : X ! Y is a morphism of projective algebraic varieties, then fK : V ! W is a regular map.

Obviously this determinant is equal (up to a sign) to the value of Rn;m at (a0 ; : : : ; an ; b0 ; : : : ; bm ). Conversely, assume that the above determinant vanishes. Then we nd a polynomial P1 (Z ) of degree n ? 1 and a polynomial Q1 (Z ) of degree m ? 1 satisfying (1). Both of them have coe cients in k. Let be a root of P (Z ) in some extension K of k. Then is a root of Q(Z )P1(Z ). This implies that Z ? divides Q(Z ) or P1 (Z ). If it divides P (Z ), we found a common root of P (Z ) and Q(Z ).