Multiplicity estimates and degeneration

作者：Fischler Stephane^*; Nakamaye Michael

来源：Manuscripta Mathematica, 2014, 143(1-2): 239-251.

DOI：10.1007/s00229-013-0635-9

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 . 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 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.

• 出版日期2014-1