Diagonal property and weak point property of higher rank divisors and certain Hilbert schemes
Arijit Mukherjee, D S Nagaraj
TL;DR
The paper introduces the diagonal property and the weak point property for ind-varieties and demonstrates their occurrence in new settings: ind-varieties of higher rank divisors with integral slope on a smooth projective curve satisfy the weak point property, while the ind-variety of $(1,n)$-divisors has the diagonal property and forms a locally complete linear ind-variety with Picard group ${\rm Pic}({\mathbf{Q}}^{1,-n}) \cong {\rm Pic}(J(C)) \oplus \mathbb{Z}$. It develops a quasi-isomorphism between ind-varieties of divisors and quotients, and computes structural data such as the Picard group. The work also analyzes Hilbert schemes associated to good partitions of a constant polynomial, proving they satisfy the diagonal property and counting them up to isomorphism by showing non-isomorphism of related multi symmetric products; this yields exact counts in terms of partition numbers. Collectively, these results connect diagonal-type properties with the geometry of ind-varieties, Hilbert schemes, and symmetric products, and provide explicit invariants like the Picard group in this broader setting.
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 and is a locally complete linear ind-variety and calculate its Picard group. Furthermore, we obtain that the Hilbert schemes of a curve associated to the good partitions of a constant polynomial satisfy the diagonal property. In the process of obtaining this, we provide the exact number of such Hilbert schemes up to isomorphism by proving that the multi symmetric products associated to two distinct partitions of a positive integer $n$ are not isomorphic.