[Paper Review] Stacks of ramified Galois covers

Fabio Tonini|arXiv (Cornell University)|Jul 3, 2013

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper studies stacks of ramified Galois covers for finite group schemes, focusing on the structure of the stack $G$-Cov for non-abelian groups like $\mu_3 \rtimes \mathbb{Z}/2\mathbb{Z}$ and $S_3$. It establishes that $G$-Cov is algebraic and finitely presented, and provides explicit geometric and cohomological invariants for $S_3$-covers of surfaces, including formulas for $K_X^2$, $\chi(\mathcal{O}_X)$, and $p_g(X)$ in terms of invariants of the base surface and associated vector bundles.

ABSTRACT

Given a finite, flat and finitely presented group scheme $G$ over some base $S$, we introduce the notion of ramified $G$-covers and study the moduli stack $G$-Cov they form. The thesis is divided in three parts. The first one concerns the case when $G$ is a diagonalizable group scheme and it essentially coincides with arxiv:1106.2347. In the second part I deal with the general case. Assuming that the base S is affine and given an $S$-scheme $T$, I interpret $G$-covers of $T$ as particolar (lax) monoidal functors from the category of finite, $G$-equivariant locally free sheaves over $S$ to the category of finite locally free sheaves over $T$, extending the classical Tannakian correspondence between $G$-torsors and strong monoidal functors as above. Using this point of view, I prove that $G$-Cov is always reducible if $G$ is a non-abelian linearly reductive group. When $G$ is constant and tame I also give a criterion to detect when a $G$-cover of a regular in codimension one, integral scheme has regular in codimension one total space in terms of the functor associated with the cover. In the last part I focus on the case $G=S_3$, prove that $S_3$-Cov has exactly two irreducible components and describe the principal one. I also describe particular open loci of $S_3$-Cov, that is particular families of $S_3$-covers, classify $S_3$-covers of regular schemes whose total space is regular and compute the invariants of $S_3$-covers of smooth surfaces.

Motivation & Objective

• To describe the algebraic and geometric structure of the stack $G$-Cov of $G$-Galois covers for finite flat group schemes $G$.
• To extend known descriptions of $G$-covers from abelian groups (e.g., $\mu_n$) to non-abelian groups, particularly $S_3$ and $\mu_3 \rtimes \mathbb{Z}/2\mathbb{Z}$.
• To provide explicit invariants—such as $K_X^2$, $\chi(\mathcal{O}_X)$, and $p_g(X)$—for $S_3$-covers of surfaces in terms of data on the base surface.
• To analyze the irreducible components and regularity properties of the stack $G$-Cov, especially in codimension one.
• To establish that the stack $G$-Cov is algebraic and finitely presented over the base scheme $S$.

Proposed method

• Uses the framework of stacks and algebraic geometry to define $G$-covers as finite, flat, $G$-invariant morphisms $f: X \to Y$ such that $f_*\mathcal{O}_X$ is locally isomorphic to the regular representation $\mathcal{O}_Y[G]$ as a comodule.
• Applies the theory of diagonalizable group schemes and $M$-graded algebras to describe $\textup{D}(M)$-covers, particularly for $M = \mathbb{Z}/3\mathbb{Z}$.
• Employs the invariant $h: |\textup{D}(M)\text{-Cov}| \to \mathbb{N}$ to classify covers by the number of generators of the associated algebra, focusing on $h \leq 2$.
• Analyzes $S_3$-covers via a triple cover construction using a vector bundle $\mathcal{F}$ and a map $\delta: \det\mathcal{F} \to \mathcal{L}$, with geometric loci defined by non-degeneracy conditions on $\alpha$, $\beta$, and $\delta$.
• Uses Riemann-Roch and sheaf cohomology to derive formulas for $K_X^2$, $\chi(\mathcal{O}_X)$, and $p_g(X)$ in terms of $K_Y$, $c_1(\mathcal{F})$, $c_2(\mathcal{F})$, and $\chi(\mathcal{O}_Y)$.
• Applies the factorization $X \to X' \to Y$ with degree 2 and 3 covers to compute canonical invariants via the adjunction formula and intersection theory on $X'$.

Experimental results

Research questions

• RQ1 How can the stack $G$-Cov be described algebraically and geometrically for non-abelian finite group schemes such as $S_3$?
• RQ2 What are the key invariants—such as $K_X^2$, $\chi(\mathcal{O}_X)$, and $p_g(X)$—for $S_3$-covers of surfaces in terms of data on the base surface?
• RQ3 What are the irreducible components of the stack $S_3$-Cov, and how do they relate to the geometry of the base and the associated vector bundles?
• RQ4 Under what conditions is a $G$-cover regular or normal in codimension one, particularly for $\mu_3 \rtimes \mathbb{Z}/2\mathbb{Z}$-covers?
• RQ5 How does the structure of the stack $G$-Cov relate to the theory of equivariant sheaves and monoidal functors for linearly reductive groups?

Key findings

• The stack $G$-Cov is algebraic and finitely presented over the base scheme $S$, and $\textup{B}G$ (the stack of $G$-torsors) is an open substack of $G$-Cov.
• For $S_3$-covers of surfaces, the formula for the square of the canonical divisor is $K_X^2 = 6K_Y^2 - 8c_1(\mathcal{F})K_Y + 4c_1(\mathcal{F})^2 - 6c_2(\mathcal{F}) - \frac{10}{3}\mu^2 - 4\mu K_Y + 2C^2 + 4CK_{X'}$, derived from factorization and Riemann-Roch.
• The Euler characteristic of the structure sheaf satisfies $\chi(\mathcal{O}_X) = \chi(\mathcal{O}_Y) + \chi(\mathcal{F}) + \chi(\mathcal{L})$, with $\chi(\mathcal{L})$ computed via $\mu$ and $c_1(\mathcal{F})$.
• The geometric genus is given by $p_g(X) = p_g(Y) + h^2(\mathcal{F})$, linking the irregularity of $X$ to the cohomology of the vector bundle $\mathcal{F}$.
• The main irreducible component $\mathcal{Z}_{S_3}$ of $S_3$-Cov is characterized by the non-degeneracy of the map $\delta: \det\mathcal{F} \to \mathcal{L}$ and the non-vanishing of $\beta: \textup{Sym}^2\mathcal{F} \to \mathcal{F}$.
• Regular $S_3$-covers are those where the associated triple cover is smooth and the base is regular in codimension one, and such covers satisfy explicit formulas for $K_X^2$ and $\chi(\mathcal{O}_X)$ in terms of invariants of $Y$ and $\mathcal{F}$.