Invariant polynomials and Mukai's models of moduli spaces of curves and K3 surfaces

Technical Summary: Invariant Polynomials and Mukai's Models of Moduli Spaces of Curves and K3 Surfaces

Problem Statement
Since the late 1980s, Shigeru Mukai has introduced birational models, denoted $M^X_{1,g}$ and $M^X_{2,g}$, for specific moduli spaces of curves and polarized K3 surfaces of genus $g \in \{7, 8, 9, 10\}$. These models are defined as Geometric Invariant Theory (GIT) quotients of Grassmannian varieties by simple linear algebraic groups. While foundational results exist regarding the stability of smooth, general objects within these models, the GIT stability of singular curves and surfaces appearing on the boundaries of these moduli spaces remains largely unknown. Previous attempts to verify the semistability of specific singular examples (such as those in genus 7) relied on classical, explicit constructions of invariant polynomials and direct evaluation. These methods were computationally prohibitive, often requiring massive parallel computing resources and significant time to evaluate even a single example.

Methodology
The paper proposes a computationally efficient approach to constructing and evaluating invariant polynomials associated with Mukai's GIT problems. The methodology proceeds in three main stages:

1. Construction of Invariant Polynomials:
   The authors analyze the decomposition of the exterior powers $\Lambda^g V_g^*$ (for curves) and $\Lambda^{g+1} V_g^*$ (for surfaces), where $V_g$ is the specific representation associated with the group $G_g$ acting on the Grassmannian. By identifying irreducible submodules within these decompositions, the authors construct invariant polynomials $F_{g,d}$ of specific degrees.

   • The strategy involves selecting irreducible submodules of small dimension and finding the lowest degree $d$ such that the symmetric power $\text{Sym}^d$ of the submodule contains a trivial $G_g$-module (an invariant).
   • For specific cases (e.g., Genus 8, Dimension 2), the authors utilize a combinatorial formula for $SL_n$-invariants expressed in terms of Gelfand-Tsetlin bases, implemented in the LieAlgebraRepresentations package for Macaulay2. This avoids the need for general-purpose, memory-intensive algorithms.

2. Localization of Singular Objects:
   The authors identify specific points in the parameter spaces (Grassmannians) corresponding to singular curves and surfaces. These include:

   • Tangent developables.
   • Cuspidal cubics with $g$-gonal symmetry.
   • Balanced K3 carpets and ribbons.
   • Graph curves associated with trivalent graphs ($\Gamma_7, \Gamma_8, \Gamma_9$).
   • Reducible surfaces and curves with generically non-reduced components.
     The coordinates for these points are derived using automorphism groups, algorithms for graph curves, and computer searches for non-reduced structures.

3. Evaluation and Verification:
   Using the constructed polynomials and the identified points, the authors evaluate $F_{g,d}$ at the parameter points representing the singular objects. If the value is non-zero, the object is proven to be GIT semistable. All calculations are performed using Macaulay2.

Key Contributions

• Algorithmic Efficiency: The paper introduces a method for evaluating invariant polynomials that is vastly more efficient than previous approaches. While prior work required days of computation on high-performance clusters (e.g., 36 hours on AWS instances), the new method constructs and evaluates polynomials for genus 7 on a standard laptop in approximately 24 hours, with subsequent evaluations being nearly instantaneous.
• Implementation: The work is implemented in the LieAlgebraRepresentations package for Macaulay2, which includes a combinatorial formula for $SL_n$-invariants in the Appendix.
• Combinatorial Formula: The Appendix provides a self-contained proof of a combinatorial formula for an $SL_n$-invariant in terms of Gelfand-Tsetlin bases, which is crucial for the efficient calculation of the genus 8 invariant.

Results
The authors prove Theorem 1.1, establishing that a wide variety of singular curves and surfaces are GIT semistable in Mukai's models for genera 7 through 10. Specifically, the paper confirms the semistability of:

• Genus 7: Tangent developables, 7-cuspidal cubics, balanced K3 carpets, balanced ribbons, and graph curves.
• Genus 8: Tangent developables, 8-cuspidal cubics, reducible surfaces with four non-reduced components, reducible curves with five non-reduced components, and graph curves.
• Genus 9: Tangent developables, 9-cuspidal cubics, balanced K3 carpets, balanced ribbons, and graph curves.
• Genus 10: Tangent developables, 10-cuspidal cubics, reducible surfaces with two non-reduced components, and reducible curves with four non-reduced components.

Notably, the genus 8 reducible surface (Example 8c) is shown to be strictly semistable, a result that required the specific invariant polynomial $F_{8,2}$ derived via the combinatorial formula, as general methods failed due to memory constraints.

Significance and Claims
The paper claims that its primary contribution is the demonstration of a scalable and efficient method for analyzing the boundaries of Mukai's moduli spaces. The authors emphasize that the "most striking features" of their results are the "length and breadth" of the list of verified semistable objects, rather than the specific geometric properties of any single item. The methodology allows for the analysis of many more examples than previously possible. The paper does not claim to solve the general problem of characterizing the entire boundary of these moduli spaces but provides a robust computational tool to verify semistability for specific, complex singular objects that were previously intractable. The work extends the findings of [25] by replacing classical, explicit calculations with a representation-theoretic approach that is both faster and more memory-efficient.