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Andrea Di Lorenzo

Publications and source records attributed to Andrea Di Lorenzo.

At least 19 recordsLinked to original sources
arXiv2026

The unbearable projectivity of being a good moduli space of Gorenstein curves of genus one with non-colliding markings

We classify all open substacks admitting a proper good moduli space inside the stack $\mathcal{G}_{1,n}$ of log-canonically polarized Gorenstein curves of genus one with $n$ distinct marked points: they are exactly the stacks $\overline{\mathcal{M}}_{1,n}(c)$ defined by Bozlee--Kuo--Neff. Restricting to the case where $c$ is pure of level $m$, we then give a numerical criterion for the projectivity of the good moduli spaces: these turn out to be asymptotically almost never projective as $n$ tends to infinity.

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arXiv2026

Degenerations of twisted maps to algebraic stacks

We give a definition of twisted map to a quotient stack with projective good moduli space, and we show that the resulting functor satisfies the existence part of the valuative criterion for properness.

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arXiv2026

Wall-crossing integral Chow rings of $\overline{\mathcal M}_{1,n}$

We compute the integral Chow rings of $\overline{\mathcal M}_{1,n}$ for $n=3,4$. For $n\leq 6$, these stacks can be obtained by a sequence of weighted blow-ups and blow-downs from a simple stack, either a weighted projective space or a Grassmannian. Our strategy consists in inductively computing all the integral Chow rings of the alternative compactifications introduced by Smyth and studied by Lekili-Polishchuk.

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arXiv2026

Stable maps to quotient stacks with a properly stable point

We compactify moduli stacks of maps from curves to a large class of quotient stacks $[W/G]$ admitting projective good moduli spaces. Our construction extends quasimap-theoretic compactifications and relies on a new birational operation for algebraic stacks, which we call an extended weighted blow-up. As applications, we construct compactifications of certain moduli spaces of fibrations in log Calabi-Yau pairs, of maps to stacks of the form $[W/\mathbb{G}_m^n]$, of maps to the GIT moduli stack of $2n$ points on $\mathbb{P}^1$, and of maps to the GIT moduli stack of plane cubics. In the appendix, we use extended weighted blow-ups to give a modular proof of a conjecture of Hassett.

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arXiv2026

A criterion for smooth weighted blow-downs

We establish a criterion for determining when a smooth Deligne-Mumford stack is a weighted blow-up. More precisely, given a smooth Deligne-Mumford stack $\mathcal{X}$ and a Cartier divisor $\mathcal{E} \subset \mathcal{X}$ such that (1) $\mathcal{E}$ is a weighted projective bundle over a smooth Deligne-Mumford stack $\mathcal{Y}$ and (2) for every $y\in\mathcal{Y}$ we have $\mathcal{O}_{\mathcal{X}}(\mathcal{E})|_{\mathcal{E}_y}\simeq \mathcal{O}_{\mathcal{E}_y}(-1)$, then there exists a contraction $\mathcal{X}\to\mathcal{Z}$ to a smooth Deligne-Mumford stack $\mathcal{Z}$. Moreover, the stack $\mathcal{X}$ can be recovered as a weighted blow-up along $\mathcal{Y}\subset \mathcal{Z}$ with exceptional divisor $\mathcal{E}$, and $\mathcal{Z}$ is a pushout in the category of algebraic stacks. As an application, we show that the moduli stack $\overline{\mathscr{M}}_{1,n}$ of stable $n$-pointed genus one curves is a weighted blow-up of the stack of pseudo-stable curves. Along the way we also prove a reconstruction result for smooth Deligne-Mumford stacks that is of independent interest.

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arXiv2026

Integral Chow rings of modular compactifications of $\mathcal{M}_{1,n\leq 6}$

For $n\leq 6$, we compute the integral Chow ring of every modular compactification of $\mathcal{M}_{1,n}$ parametrising only Gorenstein curves with smooth, distinct markings. These include the Deligne--Mumford, Schubert, and Smyth compactifications, and many more. They can all be excised from the stack of log-canonically polarised Gorenstein curves. The Chow ring of the latter admits a simple, combinatorial description, which we compute by patching along a natural stratification by core level. We further deduce that all these modular compactifications satisfy the Chow-Künneth generation property, that the cycle class map is an isomorphism, and for $n=4$ we study whether the Getzler's relation hold integrally.

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arXiv2025

Cohomological invariants of $\mathscr{M}_{3,n}$ via level structures

We show that mod $2$ cohomological invariants of the moduli stack $\mathscr{M}_{3,n}$ of smooth pointed curves of genus three contain a free module with generators in degree $0$, $2$, $3$, $4$ and $6$, formed by the invariants of the symplectic group $\mathrm{Sp}_6(2)$. We achieve this by showing that the torsor of full level two structures $\mathscr{M}_{3,n}(2) \to \mathscr{M}_{3,n}$ is versal. Along the way, we prove that the invariants of the stack of del Pezzo surfaces of degree two contain the invariants of the Weyl group $W(\mathsf{E}_7)$ and that the mod $2$ cohomology of $\mathscr{M}_{3,n}$ is non-zero in degree three. Our main result holds also for the stack $\mathscr{A}_3$ of principally polarized abelian threefolds.

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arXiv2024

The Brauer groups of moduli of genus three curves, abelian threefolds and plane curves

We compute the $\ell$-primary torsion of the Brauer group of the moduli stack of smooth curves of genus three over any field of characteristic different from two and the Brauer group of the moduli stacks of smooth plane curves of degree $d$ over any algebraically closed field of characteristic different from two, three and coprime to $d$. We achieve this result by computing the low degree cohomological invariants of these stacks. As a corollary we are additionally able to compute the $\ell$-primary torsion of the Brauer group of the moduli stack of principally polarized abelian varieties of dimension three over any field of characteristic different from two.

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arXiv2024

Effective morphisms and quotient stacks

We give a valuative criterion for when a smooth algebraic stack with a separated good moduli space is the quotient of a separated Deligne-Mumford stack by a torus. For doing so, we introduce a new class of morphisms, the so-called effective morphisms, which are a generalization of separated morphisms.

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arXiv2023

Integral Picard group of moduli of polarized K3 surfaces

We compute the integral Picard group of the moduli stack of polarized K3 surfaces of fixed degree whose singularities are at most rational double points. We also compute the integral Picard group of the stack of quasi-polarized K3 surfaces, and of the stacky period domain.

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arXiv2023

The integral Chow rings of moduli of Weierstrass fibrations

We compute the Chow rings with integral coefficients of moduli stacks of minimal Weierstrass fibrations over the projective line. For each integer $N\geq 1$, there is a moduli stack $\mathcal{W}^{\mathrm{min}}_N$ parametrizing minimal Weierstrass fibrations with fundamental invariant $N$. Following work of Miranda and Park--Schmitt, we give a quotient stack presentation for each $\mathcal{W}^{\mathrm{min}}_N$. Using these presentations and equivariant intersection theory, we determine a complete set of generators and relations for each of the Chow rings. For the cases $N=1$ (respectively, $N=2$), parametrizing rational (respectively, K3) elliptic surfaces, we give a more explicit computation of the relations.

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arXiv2023

Integral Picard group of some stacks of polarized K3 surfaces of low degree

We compute the integral Picard group of the stack $\mathcal{M}_{2l}$ of polarized K3 surfaces with at most rational double points of degree $2l=4,6,8$. We show that in this range the integral Picard group is torsion-free and that a basis is given by certain elliptic Noether-Lefschetz divisors together with the Hodge line bundle. To achieve this result, we investigate certain stacks of complete intersections and their Picard groups by means of equivariant geometry. In the end we compute an expression of the class of some Noether-Lefschetz divisors, restricted to an open substack of $\mathcal{M}_{2l}$, in terms of the basis mentioned above.

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arXiv2023

Equivariant Chow-Witt groups and moduli stacks of elliptic curves

We introduce equivariant Chow-Witt groups in order to define Chow-Witt groups of quotient stacks. We compute the Chow-Witt ring of the moduli stack of stable (resp. smooth) elliptic curves, providing a geometric interpretation of the new generators. Along the way, we also determine the Chow-Witt ring of the classifying stack of $μ_{2n}$.

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arXiv2022

Intersection theory on moduli of smooth complete intersections

We provide a general method for computing rational Chow rings of moduli of smooth complete intersections. We specialize this result in different ways: to compute the integral Picard group of the associated stack ; to obtain an explicit presentation of rational Chow rings of moduli of smooth complete intersections of codimension two; to prove old and new results on moduli of smooth curves of genus $\leq 5$ and polarized K3 surfaces of degree $\leq 8$.

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arXiv2021

A complete description of the cohomological invariants of even genus hyperelliptic curves

When the genus $g$ is even, we extend the computation of mod 2 cohomological invariants of $\mathcal{H}_g$ to non algebraically closed fields, we give an explicit functorial description of the invariants and we completely describe their multiplicative structure. In the Appendix, we show that the cohomological invariants of the compactification $\overline{\mathcal{H}}_g$ are trivial, and use our methods to give a very short proof of a result by Cornalba on the Picard group of the compactification $\overline{\mathcal{H}}_g$ and extend it to positive characteristic

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arXiv2021

Polarized twisted conics and moduli of stable curves of genus two

In this paper we introduce the stack of polarized twisted conics and we use it to give a new point of view on $\overline{\mathcal{M}}_2$. In particular, we present a new and independent approach to the computation of the integral Chow ring of $\overline{\mathcal{M}}_2$, previously determined by Eric Larson.

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