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Some recent results on the punctual Quot schemes

Indranil Biswas, Chandranandan Gangopadhyay, Ronnie Sebastian

TL;DR

This survey consolidates recent progress on the geometry of the punctual Quot scheme $\\mathrm{Quot}(E,d)$ over a smooth projective curve $C$, emphasizing three primary invariants: the $S$-fundamental group scheme, the nef cone, and the cohomology of the tangent bundle. Central to all three threads is the Hilbert-Chow map $\\phi: \\mathcal{Q}\to C^{(d)}$, whose fibers are investigated through the resolution $S_d$ to relate local fiber geometry to global invariants. The authors establish that $\\pi^S$ (and hence $\\pi^N$ and $\\pi^{\acute{e}t}$) are preserved under $\\phi$, provide explicit descriptions and bounds for the nef cone of $\\mathcal{Q}$ across genus cases, and derive detailed cohomological formulas for $T_{\\mathcal{Q}}$ via the Atiyah sequence and the Secant bundle $Sec^d(at(E))$. Together, these results illuminate how the quotient geometry of $E$ over $C$ controls deformations, automorphisms, and the birational invariants of $\\mathcal{Q}$, with consequences for moduli interpretations and intersection theory on these moduli spaces.

Abstract

Let $C$ be a smooth projective curve defined over the field of complex numbers. Let $E$ be a vector bundle on $C$, and fix an integer $d\geqslant 1$. Let $\mc Q:={\rm Quot}(E,d)$ be the Quot Scheme which parameterizes all torsion quotients of $E$ of degree $d$. In this article, we survey some recent results on various invariants of $\mc Q$.

Some recent results on the punctual Quot schemes

TL;DR

This survey consolidates recent progress on the geometry of the punctual Quot scheme over a smooth projective curve , emphasizing three primary invariants: the -fundamental group scheme, the nef cone, and the cohomology of the tangent bundle. Central to all three threads is the Hilbert-Chow map , whose fibers are investigated through the resolution to relate local fiber geometry to global invariants. The authors establish that (and hence and ) are preserved under , provide explicit descriptions and bounds for the nef cone of across genus cases, and derive detailed cohomological formulas for via the Atiyah sequence and the Secant bundle . Together, these results illuminate how the quotient geometry of over controls deformations, automorphisms, and the birational invariants of , with consequences for moduli interpretations and intersection theory on these moduli spaces.

Abstract

Let be a smooth projective curve defined over the field of complex numbers. Let be a vector bundle on , and fix an integer . Let be the Quot Scheme which parameterizes all torsion quotients of of degree . In this article, we survey some recent results on various invariants of .
Paper Structure (20 sections, 47 theorems, 123 equations)

This paper contains 20 sections, 47 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. $S$-fundamental group scheme of ${\rm Quot}(E,d)$
  3. Hilbert-Chow morphism
  4. Locus where $\phi$ is smooth
  5. The scheme $S_d$ and fibers of the Hilbert Chow map
  6. Fundamental group schemes
  7. NEF cone of ${\rm Quot}(\mathcal{O}_C^{\oplus n},d)$
  8. Nef cone of $C^{(d)}$
  9. Picard Group and Néron-Severi group of $\mathcal{Q}$
  10. Upper bound on NEF cone
  11. Nef cone in the genus 0 case
  12. Nef cone in the genus 1 case
  13. Nef cone in the genus $g\geqslant 2$ case
  14. Automorphisms and Deformations of $\mathcal{Q}$
  15. Atiyah sequence
  16. ...and 5 more sections

Key Result

Lemma 2.2.2

The map $\phi\,:\,\mathcal{Q}\,\longrightarrow \,C^{(d)}$ is smooth at $q$ (corresponding to the quotient in equation thick-points).

Theorems & Definitions (80)

  • Lemma 2.2.2
  • Proposition 2.2.3
  • proof
  • Remark 2.3.9
  • Lemma 2.3.12
  • Proposition 2.3.13
  • Lemma 2.3.14
  • Corollary 2.3.15
  • proof
  • Definition 2.4.1
  • ...and 70 more