Research

Abstract: We investigate monodromy groups arising in enumerative geometry, with a particular focus on how these groups are influenced by prescribed symmetries. To study these phenomena effectively, we work in the framework of moduli stacks rather than moduli spaces. This perspective proves broadly useful for understanding and constructing monodromy. We illustrate these ideas through several examples, with special attention to the 27 lines on a cubic surface, assuming the surface admits a given symmetry group.

Abstract: This paper is the second in a series devoted to describing the integral Chow ring of the moduli stacks RH_g of hyperelliptic Prym pairs. For fixed genus g, the stack RH_g is the disjoint union of $\lfloor (g+1)/2 \rfloor$ components RH_g^n for $n = 1,...,\lfloor (g+1)/2 \rfloor$. In this paper, we compute the integral Chow rings of the components RH_g^{(g+1)/2} for odd g. Along the way, we also determine the integral Chow ring of the moduli stack of unordered pairs of two divisors on the projective line of the same even degree.

Abstract: This paper is the first in a series dedicated to computing the integral Chow rings of the moduli stacks of Prym pairs. In this work, we compute the Chow ring for Prym pairs arising from a single pair of Weierstrass points and from at most (g-1)/2 pairs when the genus g of the curve is odd.

Abstract: We study the integral Chow ring of the stack of n-pointed smooth hyperelliptic curves of genus g. We compute it for n=1,2 completely, while for 3<=n<=2g+2 we compute it up to the additive order of a signle class in degree 2. We obtain partial results also for n=2g+3. In particular, taking g=2, our results hold for the Chow ring of M_{2,n} for 1<=n<=7.

Abstract: We prove that the relative log canonical algebra of a semi-dlt pair $(X,\Delta_X)$ projective over a scheme $T$ is finitely generated, equivalently, that $(X,\Delta_X)$ admits a relative stable model, provided that the normalization admits a log canonical model over $T$, that $(X,\Delta_X)$ admits a stable model over an open dense subscheme $T^0\subset T$ whose exceptional locus does not contain any stratum of the conductor, and that the log centers contained in the conductor have images meeting $T^0$. Our main contribution is a direct proof that avoids Koll\'ar's gluing theory entirely.

As a consequence, we recover demi-normal versions of results of Hacon--Xu and Birkar. Furthermore, we derive the existence of certain MMP steps for slc pairs, recovering results of Ambro and Koll\'ar. Finally, these results lay the groundwork for a streamlined proof of the properness of the Koll\'ar--Shepherd-Barron--Alexeev moduli space of stable pairs, to be completed in forthcoming work.

Abstract: At the genesis of Galois theory at the end of the 19th century, enumerative geometers were concerned with the \emph{solvability} of various problems: for instance whether one can express lines on a cubic surface in radicals in terms of the coefficients that define it. By work of Hermite and a celebrated theorem of Harris, this is equivalent to determining whether the monodromy group of a certain finite \'etale cover is solvable. This problem, as well as the related problem of computing bitangents to plane quartics, are both unsolvable. When one restricts to the locus of $G$-symmetric cubic surfaces, monodromy will drop and can become solvable. In this setting there are four flavors of monodromy one can consider: the one from restricting the classical cover, the one over the GIT quotient after modding out by projective transformations, and two related notions of ``monodromy'' coming from the stacky cover of $G$-symmetric lines on $G$-symmetric cubic surfaces --- unlike in the non-symmetric setting these are all different. In this paper we characterize the relationships between all four notions of monodromy in the general setting of quotient stacks, and use these connections to compute all these monodromy groups for all 11 possible automorphism groups of smooth cubic surfaces, as well as for all 12 possible automorphism groups for smooth planar quartics. We demonstrate that, in the presence of \emph{any symmetry} whatsoever, the problem of solving for lines on cubic surfaces or bitangents to plane quartics is solvable.

Abstract: We compute the K-theory of weighted blowups of smooth stacks satisfying the resolution property along smooth centers. As an application, we determine the K-theory of the stack of stable genus 1 curves with 2 marked points. Furthermore, we express the Lambda polynomial of the tangent complex of the blowup morphism in terms of the blowup data. Along the way, we extend the operational K-theory from torus actions to actions of smooth affine algebraic groups.

Abstract: Let \mathcal{X} be an algebraic stack admitting a moduli space \mathcal{X}_{\mathrm{mod}}. We study the factorizations of the moduli space morphism \mathcal{X}\rightarrow\mathcal{X}_{\mathrm{mod}} to construct intermediate stacks that simplify the stacky structure of \mathcal{X} while retaining more structural information than \mathcal{X}_{\mathrm{mod}}. Under mild assumptions, we prove the existence of a universal morphism from \mathcal{X} to stacks satisfying well-behaved `modular properties' (such as being Deligne-Mumford, having finite inertia, or being uniformizable), and show that this universal map is itself an adequate moduli space morphism. We achieve this by proving that ascending chains of adequate moduli space morphisms from a Noetherian stack stabilize if they are cohomologically affine or with target Deligne-Mumford stacks. Finally, we demonstrate that stabilization completely fails for general adequate moduli space morphisms. We construct a simple Noetherian, Deligne-Mumford stack admitting an infinite, non-stabilizing chain of adequate moduli space morphisms, whose limit is a non-algebraic fpqc stack.

Abstract: This paper is the third and final part of a series devoted to the description of the integral Chow rings of the moduli stacks of hyperelliptic Prym pairs. For a fixed genus $g$, there are two natural stacks, $\RH_g$ and $\wRH_g$, parametrizing hyperelliptic Prym pairs, with the former being the $\mu_2$-rigidification of the latter. Both decompose as the disjoint union of $\lfloor (g+1)/2 \rfloor$ components, denoted $\mathcal{RH}_g^n$ and $\wRH_g^n$ for $n = 1, \ldots, \lfloor (g+1)/2 \rfloor$. In this paper we present quotient stack descriptions of the components $\mathcal{RH}_g^n$ for even $g$ and compute their integral Chow rings, thereby completing the computation for all irreducible components of $\RH_g$. In addition, we give quotient stack presentations for all irreducible components of $\widetilde{\mathcal{RH}}_g$ and determine when the rigidification map $\widetilde{\mathcal{RH}}_g^n \to \RH_g^n$ is a root gerbe.  We then use this to compute the Chow rings of $\wRH_g^n$ for all $g$ and $n$, with the sole exception of the case where $g$ is odd and $n=(g+1)/2$.

Finally, in the appendix, we discuss $G$-gerbes induced by an homomorphism of abelian groups $H \to G$ and an $H$-gerbe.

Abstract: We study the stack H_{r,g,n} of n-pointed smooth cyclic covers of degree r between smooth curves of genus g and the projective line. We give two presentations of an open substack of H_{r,g,n} as a quotient stack, and we study its complement. Using this, we compute the integral Picard group of H_{r,g,n}. Moreover, we obtain a very explicit description of the generators of the Picard group, which have evident geometric meaning. As a corollary of the computation, we get the integral Picard group of the stack H_{g,n} of n-pointed hyperelliptic curves of genus g. Finally, taking g=2 and recalling that H_{2,n}=M_{2,n}, we obtain Pic(M_{2,n}).