Riemann-Roch type inequalities. (English) Zbl 0538.14006

The aim of the paper is to give estimates for \(\ell(rX)\) depending only on the two highest coefficients of the Hilbert polynomial of X. Here X is a semi-ample Cartier divisor on a normal projective variety (semi-ample divisor means that some multiple of it has no base points and defines a birational map). The precise result is the following: For every n, there is a polynomial Q(x,y,z) with positive rational coefficients of degree in z at most n-1, satisfying the property: for any normal projective variety V of dimension n, any semi-ample Cartier divisor on V, and any integer r one has \(| \ell(rX)-(d/n!)r^ n| \leq Q(d,\xi,r),\) where \(d=X^{(n)}\) and \(\xi =I(K_ V,X^{(n-1)})\). - The paper starts with the particular case of ample divisors on smooth projective varieties, which is easier to prove, and ends with a proof that the family of smooth polarized varieties (V,X) having bounded intersection numbers \(X^{(n)}\), \((K_ V\cdot X^{(n-1)})\) is a bounded family.
Reviewer: M.Stoia


14C40 Riemann-Roch theorems
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
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