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A geometric effective Nullstellensatz. (English) Zbl 0944.14003

The aim of the paper is to prove a geometric theorem on the effective Nullstellensatz, extending and giving more geometric light on previous works of algebraic nature. Let \(X\) be a smooth complex projective variety, \(\dim X=n\), \(D_1,\dots, D_m\in|D|\) effective divisors on \(X\) lying in a given linear series. Let also \(L= {\mathcal O}_X(D)\), \(s_j\in \Gamma(X,L)\) be the sections defining \(D_j\), \(B\) the scheme-theoretic intersection \(D_1\cap\dots\cap D_m\) and \(J\) its ideal sheaf, \(Z:= B_{\text{red}}\). One has \(Z= Z_1\cup \dots\cup Z_t\), where \(Z_i\) are the supports of the irreducible components of \(\mathbb{P} (C_{B/X})\), the projectivized normal cone of \(B\) in \(X\), and \(r_i\) the multiplicity of the corresponding component of the exceptional divisor in the blowing-up of \(X\) along \(B\). Let \(I_{Z_i}\) be the ideal sheaf of \(Z_i\) and \(I_{Z_i}^{(r)}\) its \(r\)-th symbolic power. Suppose that \(L\) is ample. Then:
(a) \(\sum r_i\deg_L (Z_i)\leq \deg_L(X)= \int_X c_1(L)^n\);
(b) \(I_{Z_1}^{(nr_1)}\cap \dots\cap I_{Z_t}^{(nr_t)} \subseteq J\);
(c) If \(K_X\) is a canonical divisor of \(X\), \(A\) a divisor on \(X\) such that \(A- (n+1)D\) is ample and if \(s\in \Gamma(X, {\mathcal O}_X(K_X+ A))\) is a section vanishing to order \((n+1)r_i\) at the general point of each \(Z_i\), then there are \(h_j\in \Gamma(X,{\mathcal O}_X (K_X+ A-D_j))\) such that \(s= \sum s_j h_j\).

MSC:

14A05 Relevant commutative algebra
14C20 Divisors, linear systems, invertible sheaves
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