The Seshadri Constants of Tangent Sheaves on Toric Varieties

TL;DR

This paper characterizes when the Seshadri constant of the tangent sheaf on a toric variety is positive, linking it to the combinatorial structure of the fan, and confirms a conjecture in a special case.

Contribution

It provides a necessary and sufficient condition for the positivity of the Seshadri constant of tangent sheaves on toric varieties, and confirms a conjecture for certain smooth projective cases.

Findings

Positivity of Seshadri constant relates to fan combinatorics.

Characterization of toric varieties with positive tangent sheaf Seshadri constants.

Confirmation of a conjecture for some smooth projective toric varieties.

Abstract

In this paper, we investigate the Seshadri constant $\varepsilon(X,T_X;p)$ of the tangent sheaf $T_X$ on a complete $\mathbb Q$-factorial toric variety $X$. We show that $\varepsilon(X,T_X;1)>0$ if and only if the following statement holds true: if $a_1v_1+\cdots +a_kv_k=0$ where $a_i$'s are positive real numbers and $v_i$'s are primitive generators of some rays in the fan $\Delta$ that defines $X$, then $k\geq \dim X+1$. Based on the result, we show that a smooth projective toric variety $X$ with $\varepsilon(X,T_X;p)>0$ for some $p\in X$ is isomorphic to the projective space, confirming a special case of the conjecture proposed by M. Fulger and T. Murayama.