Intersections on Non-singular Varieties

If Y is a non-singular variety, the diagonal imbedding δ of Y in Y x Y is a regular imbedding. For x, y ∈ A * Y, the product x·y ∈ A * Y is defined by the formula $$ x \cdot y = {\delta ^ * }\left( {x \times y} \right) $$ Setting $$ {A^P}Y = {A_{N - P}}Y $$ , n = dim Y, this product makes A * Y into a commutative, graded, ring, with unit [Y]. If f: X → Y is a morphism, with Y non-singular, the graph morphism γ f from X to X x Y is a regular imbedding. For x ∈ A * X, y ∈ A * Y, define $$ \chi { \cdot _f}y = \gamma \frac{ * }{f}\left( {x \times y} \right) \in {A_ * }{\rm X} $$ This product makes A * X into a graded module over A*Y. If X is also nonsingular, setting $$ {f^ * }\left( y \right) = \left[ {\rm X} \right]{ \cdot _f}y $$ defines a homomorphism f*: A* Y → A*X of graded rings. Using the refined operation $$ y\frac{!}{f} $$ in place of $$ y\frac{ * }{f},x{ \cdot _f}y $$ has a canonical refinement in $$ {A_ * }\left( {\left| x \right| \cap {f^{ - 1}}\left( {\left| y \right|} \right)} \right) $$ In particular, if V and W are subvarieties of a non-singular variety Y, the intersection class V·W is defined in Am, (V∩W)is defined in $$ {A_m}\left( {V \cap W} \right) $$ m = dim V +dim W - dim Y. Any m-dimensional irreducible component Z of V∩W W has a coefficient in V·W, called the intersection multiplicity, and denoted i(Z, V·W; Y). The expected properties of these intersection products and multiplicities follow readily from the general properties proved in Chaps. 6 and 7. Bézout’s theorem, in its simplest form, states that $$ {A^ * }\left( {{\mathbb{P}^n}} \right) \cong \mathbb{Z}\left[ h \right]/\left( {{h^{n + 1}}} \right) $$ where h is the class of a hyperplane. A deeper analysis of intersections on projective space will be given in Chap. 12.