Abstract
In this paper we prove a multiplicity estimate, which is best possible up to a multiplicative constant, in which the set of points is connected to an action of SL2(Z) on the torus \({{\bf G}_m^2({\bf C})}\). This result is motivated by the construction, due to Roy, of a non-trivial auxiliary function that could be used to study the points on the Grassmannian whose coordinates are logarithms of algebraic numbers: to make use of this construction, only a zero estimate connected to the action of GL m (Z) on \({\wedge^k {\bf C}^m}\) is missing. The result we prove is essentially analogous to it. The proof is based on the fact that a zero (or multiplicity) estimate can be derived from a lower bound for a Seshadri constant. Then a degeneration argument is used: inside a family any such lower bound holds on an open subset, so proving it for sufficiently many special cases yields it for almost all cases.
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Fischler, S., Nakamaye, M. Multiplicity estimates and degeneration. manuscripta math. 143, 239–251 (2014). https://doi.org/10.1007/s00229-013-0635-9
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DOI: https://doi.org/10.1007/s00229-013-0635-9