A Schmidt-Nochka Theorem for closed subschemes in subgeneral position

TL;DR

This paper generalizes Schmidt's subspace theorem to weighted sums of closed subschemes in subgeneral position, extending classical inequalities and applying algebraic geometry techniques to improve bounds in Nevanlinna theory.

Contribution

It introduces a new generalized theorem for weighted sums of subschemes in subgeneral position, unifying Schmidt's theorem with Nochka's inequalities, and extends these results to hypersurfaces and Nevanlinna theory.

Findings

Extended inequalities of Nochka and Ru-Wong to hypersurfaces in subgeneral position.

Proved a sharp result in dimensions 2 and 3, near-sharp bounds in all dimensions.

Generalized the Second Main Theorem and Nochka's theorem in Nevanlinna theory.

Abstract

In previous work, the authors established a generalized version of Schmidt's subspace theorem for closed subschemes in general position in terms of Seshadri constants. We extend our theorem to weighted sums involving closed subschemes in subgeneral position, providing a joint generalization of Schmidt's theorem with seminal inequalities of Nochka. A key aspect of the proof is the use of a lower bound for Seshadri constants of intersections from algebraic geometry, as well as a generalized Chebyshev inequality. As an application, we extend inequalities of Nochka and Ru-Wong from hyperplanes in $m$-subgeneral position to hypersurfaces in $m$-subgeneral position in projective space, proving a sharp result in dimensions $2$ and $3$, and coming within a factor of $3/2$ of a sharp inequality in all dimensions. We state analogous results in Nevanlinna theory generalizing the Second Main Theorem and Nochka's theorem (Cartan's conjecture).