Effective divisors and Newton-Okounkov bodies of Hilbert schemes of points on toric surfaces

TL;DR

This paper computes the Newton-Okounkov body for Hilbert schemes of points on certain surfaces, providing bounds and conjectures that deepen understanding of their geometric properties and effective cones.

Contribution

It introduces an explicit computation of the Newton-Okounkov body for Hilbert schemes on a2^2 and proposes conjectural exact descriptions for key toric surfaces.

Findings

Computed the Newton-Okounkov body for a2^2 Hilbert scheme.

Provided upper bounds for Newton-Okounkov bodies on smooth toric surfaces.

Suggested conjectural equalities for specific surfaces like a2^2, a2^1a2^1, and Hirzebruch surfaces.

Abstract

We compute the (unbounded) Newton-Okounkov body of the Hilbert scheme of points on $\mathbb C^2$. We obtain an upper bound for the Newton-Okounkov body of the Hilbert scheme of points on any smooth toric surface. We conjecture that this upper bound coincides with the exact Newton-Okounkov body for the Hilbert schemes of points on $\mathbb P^2$, $\mathbb P^1\times\mathbb P^1$, and Hirzebruch surfaces. These results imply upper bounds for the effective cones of these Hilbert schemes, which are also conjecturally sharp in the above cases.