INTRODUCTION TO ALGEBRAIC GEOMETRY Note del corso di ...

56 Mezzetti((a 1 , . . ., a n ), λ) ∈ Y 0 × A 1 → (λ, λa 1 , . . ., λa n ) ∈ C(Y 0 ).Therefore dim C(Y 0 ) = dim(Y 0 × A 1 ) = dimY 0 + 1. To conclude, it is enough toremark that dim Y = dimY 0 and dim C(Y ) = dim C(Y 0 ) = dim S(Y ). □11.4. Corollaries.1. If X, Y ⊂ P 2 are projective curves over an algebraically closed field, thenX ∩ Y ≠ ∅.2. P 1 × P 1 is not isomorphic to P 2 .Proof. 1. is a straightforward application of Theorem 11.2. To prove 2., assumeby contradiction that φ : P 1 × P 1 → P 2 is an isomorphism. If L, L ′ are skew lineson P 1 × P 1 , then φ(L), φ(L ′ ) are rational disjoint curves of P 2 , which contradicts1.If X, Y ⊂ P n are varieties of dimensions r, s, then r + s − n is called theexpected dimension of X ∩ Y . If all irreducible components Z of X ∩ Y have theexpected dimension, then we say that the intersection X ∩ Y is proper or that Xand Y intersect properly.For example, two plane projective curves X, Y intersect properly if they don’thave any common irreducible component. In this case, it is possible to previewthe number of points of intersections. Precisely, it is possible to associate to everypoint P ∈ X ∩ Y a number i(P), called the multiplicity of intersection of X andY at P, in such a way that ∑ P ∈X∩Y i(P) = dd′ , where d is the degree of X andd ′ is the degree of Y . This result is known as Theorem of Bézout, and is the firstresult of the branch of algebraic geometry called Intersection Theory. For a proofof the Theorem of Bézout, see for instance the classical book of Walker [8], or thebook of Fulton on Algebraic Curves [5].Let X be a closed subvariety of P n (resp. of A n ) of codimension r. X is calleda complete intersection if I h (X) (resp. I(X)) is generated by r polynomials.Hence, if X is a complete intersection of codimension r, then X is certainly theintersection of r hypersurfaces. Conversely, if X is intersection of r hypersurfaces,then, by Theorem 11.1, using induction, we deduce that dimX ≥ n − r; alsoassuming equality, we cannot conclude that X is a complete intersection, butsimply that I(X) is the radical of an ideal generated by r polynomials.11.5. Example. Let X ⊂ P 3 be the skew cubic. The homogeneous ideal of Xcontains the three polynomials F 1 , F 2 , F 3 , the 2 × 2–minors of the matrixM =( )x0 x 1 x 2,x 1 x 2 x 3which are linearly independent polynomials of degree 2. Note that I h (X) does notcontain any linear polynomial, because X is not contained in any hyperplane, andthat the homogeneous component of minimal degree 2 of I h (X) is a vector space