Diagonal property and weak point property of higher rank divisors and certain Hilbert schemes

TL;DR

This paper introduces new geometric properties called the diagonal and weak point properties for ind-varieties, demonstrating these properties for higher rank divisors and certain Hilbert schemes, and establishing bounds on their classifications.

Contribution

It defines the diagonal and weak point properties for ind-varieties and proves these properties for higher rank divisors and specific Hilbert schemes, including bounds on their classification.

Findings

Ind-varieties of higher rank divisors have the weak point property.

The ind-variety of (1,n)-divisors has the diagonal property.

An upper bound on the number of Hilbert schemes up to isomorphism is established, and it is shown to be sharp for genus zero curves.

Abstract

In this paper, we introduce the notion of the diagonal property and the weak point property for an ind-variety. We prove that the ind-varieties of higher rank divisors of integral slopes on a smooth projective curve have the weak point property. Moreover, we show that the ind-variety of $(1,n)$-divisors has the diagonal property. Furthermore, we obtain that the Hilbert schemes associated to the good partitions of a constant polynomial satisfy the diagonal property. On the process of obtaining this, we provide an upper bound on the number of such Hilbert schemes up to isomorphism. Furthermore, we prove that the obtained upper bound is attained in case of genus zero curves and hence conclude that the bound is sharp.