Multiplicity lemmas and intersections
This paper studies one of the postulation problems encountered in transcendence theory, where the ambient variety is a smooth compactification \(X\) of a complex commutative algebraic group \(G\), and the zero-dimensional subscheme \(Y\) is the orbit under a finite set of translations of the origin of \(G\), thickened along a subspace of \(\text{Lie} (G)\). The author gets rid of an unnatural constant which occurred in the fundamental upper bound obtained by \textit{P. Philippon} [Bull. Soc. Math. Fr. 114, 355-383 (1986; Zbl 0617.14001)] for the minimal degree of the effective divisors on \(G\) containing \(Y\). The new point in the proof is the appeal to intersection theory, combined with an explicit description of \(\text{Pic} (X)\), and standard properties of Cohen-Macaulay schemes. Remarks: Another type of improvement of the constants in his upper bound has recently been obtained by \textit{P. Philippon} [Rocky Mt. J. Math. 26, 1069-1088 (1996)] while another approach to multiplicity estimates, still based on intersection theory, is given by \textit{M. Nakamaye} [Bull. Soc. Math. Fr. 123, No. 2, 155-188 (1995; Zbl 0841.11037)].
scientific article; zbMATH DE number 790580