INTRODUCTION TO ALGEBRAIC GEOMETRY Note del corso di ...

12 Mezzetti3.1. Proposition. Let K be an algebraically closed field. Let Z ⊂ P n be aprojective hypersurface of degree d. Then a line of P n , not contained in Z, meetsZ at exactly d points, counting multiplicities.Proof. Let G be the reduced equation of Z and L ⊂ P n be any line.We fix two points on L: A = [a 0 , . . ., a n ], B = [b 0 , . . ., b n ]. So L admitsparametric equations of the form⎧x 0 = λa 0 + µb 0⎪⎨x 1 = λa 1 + µb 1⎪⎩ . . .x n = λa n + µb nThe points of Z ∩L are obtained from the homogeneous pairs [λ, µ] which aresolutions of the equation G(λa 0 + µb 0 , . . ., λa n + µb n ) = 0. If L ⊂ Z, then thisequation is identical. Otherwise, G(λa 0 + µb 0 , . . ., λa n + µb n ) is a non-zero homogeneouspolynomial of degree d in two variables. Being K algebraically closed, itcan be factorized in linear factors:G(λa 0 + µb 0 , . . ., λa n + µb n ) = (µ 1 λ − λ 1 µ) d 1(µ 2 λ − λ 2 µ) d 2. . .(µ r λ − λ r µ) d rwith d 1 + d 2 + . . . + d r = d. Every factor corresponds to a point in Z ∩ L, to becounted with the same multiplicity as the factor.□If K is not algebraically closed, considering the algebraic closure of K andusing Proposition 3.1, we get that d is un upper bound on the number of pointsof Z ∩ L.c) Affine and projective subspaces.The subspaces introduced in §1, both in the affine and in the projective case,are examples of algebraic sets.d) Product of affine spaces.Let A n , A m be two affine spaces over the field K. The cartesian productA n ×A m is the set of pairs (P, Q), P ∈ A n , Q ∈ A m : it is in natural bijection withA n+m via the mapφ : A n × A m −→ A n+msuch that φ((a 1 , . . ., a n ), (b 1 , . . ., b m )) = (a 1 , . . ., a n , b 1 , . . ., b m ).From now on we will always identify A n × A m with A n+m . We get twotopologies on A n × A m : the Zariski topology and the product topology.3.1. Proposition. The Zariski topology is strictly finer than the product topology.Proof. If X = V (α) ⊂ A n , α ⊂ K[x 1 , . . ., x n ] and Y = V (β) ⊂ A m , β ⊂K[y 1 , . . ., y m ], then X × Y ⊂ A n × A m is Zariski closed, precisely X × Y =