Positivity

We have constructed intersection classes by intersecting a cone C in a normal bundle N with the zero-section. If Σmi [Ci] is the cycle of C, the intersection class has a corresponding decomposition into Σmiαi, $$ {\alpha _i} \in {A_*}\left( {{Z_i}} \right) $$ , $$ {Z_i} = Supp({C_i}) $$ . If the bundle N is suitably positive, one can deduce corresponding positivity of the intersection classes, even if the intersections are not proper. Assume for simplicity that the restriction N i of N to Z i; is generated by its sections. Then αi is represented by a non-negative cycle. If N i; is also ample, ai is represented by a positive cycle. If is generated by its sections, for an ample line bundle L, then using L to compute degrees, the degree of αi is bounded below by the degree of Z i . For intersections on a non-singular variety X, the positivity of its tangent bundle will imply corresponding positivity for all intersection classes on X. For $$ X = I{P^N} $$ , V, subvarieties, a refined Bézout’s theorem follows: $$ {V_1}{._{...}}.{V_r} = \sum\nolimits_{{m_i}{\alpha _i}} {} $$ , $$ \mathop \prod \limits_{j = 1}^r \deg \left( {{V_J}} \right) = \sum\limits_{} {{m_i}\deg } \left( {{\alpha _i}} \right) \geqslant \sum\limits_{} {{m_i}\deg } \left( {{Z_i}} \right) $$ where the Zi are the distinguished varieties; all irreducible components of ∩j Vj are included among the Zj. There are also applications to intersection multiplicities. For example, if…, Vr meet properly at a non-singular point P of an n-dimensional variety X, and $$ \tilde V \subset \tilde X $$ are the blow-ups at P, then $$ i\left( {p,{V_1}{._{...}}.{V_r};} \right) = \prod\limits_{j = 1}^r {{e_p}} \left( {{V_j}} \right) + \deg \left( {{{\tilde V}_1}{._{...}}.{{\tilde V}_1}} \right) $$ Here the intersection class is in A o (E), $$ E \cong I{P^{n - 1}} $$ the exceptional divisor. The degree of is always non-negative, and one has lower bounds as in the refined Bézout’s theorem, e.g. $$ \deg \left( {{{\tilde V}_1}{._{...}}.\tilde V} \right) \geqslant \sum\limits_{i = 1}^s {\deg \left( {{W_i}} \right)} $$ , where are the irreducible components of the intersection ∩j P(CpVj) of the projective tangent cones. Such positivity is noteworthy since the $$ {\widetilde V_i}$$ may have excess intersections, and general intersections on $$ \widetilde X$$ can be negative. There are similar inequalities for proper intersections of divisors on a possibly singular variety. Notation. Acycle Σni [Vi] on a scheme X is non-negative if each ni is non-negative, and positive if, in addition, at least one ni is positive. Let Ak ≥X (resp.A) k + X) denote the set of classes in Ak X which can be represented by non-negative (resp.positive) cycles. Thus $$ A_k^ \geqslant X = A_k^ + X \cup \{ 0\}.$$ , Both sets are clesed under addition. Let L be a line bundle on a complete scheme X. For a k-cycle or cycle class αon X,he L-degree of α,denoted degL(α), is defined by $$ de{s_L}(\alpha ) = \int\limits_X {{c_1}} {(L)^k} \cap \alpha . $$ . If V a subvariety of X,the L-degree of V, degL(L),is defined by $$ {\deg _L}(V) = {\deg _L}([V]) = \int\limits_x {{c_1}} {(L)^{\dim V}} \cap [V].$$ .