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Seshadri constants and interpolation on commutative algebraic groups. (Constantes de Seshadri et interpolation dans les groupes algèbriques commutatifs.) (English. French summary) Zbl 1330.14076
Interpolation estimates occur in transcendence methods as a tool for giving upper bounds for the degree of polynomials taking given values (which can involve multiplicities) at a finite number of points. They provide estimates, related to auxiliary analytic functionals, which can be compared with zero or multiplicity estimates. Zero (and multiplicity) estimates are related to auxiliary analytic functions and provide lower bounds for the degree of polynomials vanishing (with multiplicities) at a finite number of points. While the theory of zero estimates and multiplicity estimates on algebraic groups has reached a satisfactory level, only few interpolation estimates were known so far: D. W. Masser [Prog. Math. 31, 151–171 (1983; Zbl 0579.14038)], S. Fischler [Compos. Math. 141, No. 4, 907–925 (2005; Zbl 1080.14054)].
In the present paper, the authors develop the theory of interpolation estimates on a commutative algebraic group endowed with the Serre compactification and reach the same level of accuracy as the theory of multiplicity estimates. They relate the algebraic subgroups responsible for a possible defect in the interpolation estimate to a Seshadri exceptional subvariety, in a way similar to what was done for multiplicity estimates [M. Nakamaye and N. Ratazzi, Math. Z. 259, No. 4, 915–933 (2008; Zbl 1156.11027)]. This paper contains results which have their own independent interest related with Seshadri’s constant and Seshadri’s exceptional varieties.

14L10 Group varieties
14C20 Divisors, linear systems, invertible sheaves
11J95 Results involving abelian varieties
14L40 Other algebraic groups (geometric aspects)
Full Text: DOI
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