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Sheaf stable pairs on projective surfaces and birational geometry

Caucher Birkar, Jia Jia, Artan Sheshmani, Chengxi Wang

TL;DR

The paper develops a birational perspective on moduli spaces of higher-rank sheaf stable pairs on smooth projective surfaces by linking them to stable minimal models through log-canonical models and Maruyama transformations. It constructs and analyzes the Hilbert–Chow morphism to CDiv(Z), describing its fibres in the Picard number one setting and revealing a rich fibre structure governed by projective bundle and Quot-scheme components. In particular, it gives explicit descriptions for rank 2 and negative-curve scenarios: when C^2=−d, the moduli M_Z(2,2C,−2d) has two smooth components intersecting along Γ ≅ P^d×P^1, with detailed results for d=1 and d≥2. The work highlights a deep interplay between enumerative invariant theory, birational geometry, and moduli of decorated sheaves on surfaces, suggesting broader connections between enumerative and birational facets of geometry.

Abstract

We study moduli space of higher rank marginally stable pairs (E,s:= (s_1,..., s_r)) consisting of torsion free coherent sheaf E of rank r and r sections (s_1,..., s_r) on a smooth projective surface. Having fixed the Chern character of E, the resulting moduli space is isomorphic to some subscheme of the Quot-scheme parametrising quotient sheaves of appropriate Chern character. We establish a connection between moduli space of higher rank stable pairs and stable minimal models induced by the sheaf E and sections s_i and the relative lc model of base surface, and use birational geometry of minimal models to analyse in detail the components of the fibre of the Hilbert-Chow morphism from the moduli space to the Hilbert scheme of effective Cartier divisors on the base surface.

Sheaf stable pairs on projective surfaces and birational geometry

TL;DR

The paper develops a birational perspective on moduli spaces of higher-rank sheaf stable pairs on smooth projective surfaces by linking them to stable minimal models through log-canonical models and Maruyama transformations. It constructs and analyzes the Hilbert–Chow morphism to CDiv(Z), describing its fibres in the Picard number one setting and revealing a rich fibre structure governed by projective bundle and Quot-scheme components. In particular, it gives explicit descriptions for rank 2 and negative-curve scenarios: when C^2=−d, the moduli M_Z(2,2C,−2d) has two smooth components intersecting along Γ ≅ P^d×P^1, with detailed results for d=1 and d≥2. The work highlights a deep interplay between enumerative invariant theory, birational geometry, and moduli of decorated sheaves on surfaces, suggesting broader connections between enumerative and birational facets of geometry.

Abstract

We study moduli space of higher rank marginally stable pairs (E,s:= (s_1,..., s_r)) consisting of torsion free coherent sheaf E of rank r and r sections (s_1,..., s_r) on a smooth projective surface. Having fixed the Chern character of E, the resulting moduli space is isomorphic to some subscheme of the Quot-scheme parametrising quotient sheaves of appropriate Chern character. We establish a connection between moduli space of higher rank stable pairs and stable minimal models induced by the sheaf E and sections s_i and the relative lc model of base surface, and use birational geometry of minimal models to analyse in detail the components of the fibre of the Hilbert-Chow morphism from the moduli space to the Hilbert scheme of effective Cartier divisors on the base surface.
Paper Structure (16 sections, 40 theorems, 145 equations)

This paper contains 16 sections, 40 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Contractions
  4. Pairs and singularities
  5. Types of models
  6. Chern character
  7. Support of a sheaf
  8. Cross-sections of ruled surfaces
  9. Sheaf stable pairs revisit
  10. Geometric treatment
  11. Smooth projective surfaces of Picard number $1$
  12. Negative curves
  13. Study $M_f$
  14. Study $M_f'$
  15. $d=1$
  16. ...and 1 more sections

Key Result

Theorem 1.3

Let $Z$ be a smooth projective surface of Picard number one and let $H$ be an ample generator of $\mathop{\mathrm{NS}}\nolimits(Z)$. Consider an element $[\mathcal{E},s]\in M_Z(r,dH,d^2H^2/2)$, i.e., a sheaf stable pair $\mathcal{E},s$ with rank $\mathop{\mathrm{ch}}\nolimits(\mathcal{E})=(r,dH,d^2H whose fiber over any point $[n_iC_i]$ is isomorphic to $\mathbb{P}^{r-1}$.

Theorems & Definitions (93)

  • Definition 1.1: birkar2024sheaf
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4: =thm:moduli_space_d1
  • Theorem 1.5: =thm:main_negative_curve
  • Definition 2.1: birkar2024sheaf
  • Lemma 2.2: friedman1998algebraic
  • Definition 2.3: cf. huybrechts2010geometry
  • Definition 2.4: huybrechts2010geometry
  • Definition 2.5: stacks-project
  • ...and 83 more