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Michele Pernice

Publications and source records attributed to Michele Pernice.

15 recordsLinked to original sources
arXiv2026

K-theory of Weighted Blowups

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 generalize the Anderson-Payne construction of equivariant operational K-theory from torus actions to actions of smooth affine algebraic groups.

math.AG↗
arXiv2026

Obstructions to the existence of good moduli spaces of $A_r$-stable curves

We study obstructions to the existence of separated good moduli spaces for open substacks of the moduli stack $\mathcal{M}_{g,n}^r$ of $A_r$-stable curves. Our approach is based on an analysis of families of curves over $\Theta_R$ and $\overline{\text{ST}}_R$, building on prior work on the local geometry of $\mathcal{M}_{g,n}^r$. We prove that $\mathcal{M}_{g,n}^r$ is neither $\Theta$- nor $\textsf{S}$-complete. We then construct an open substack $\mathcal{U}_{g,n}^r \subset \mathcal{M}_{g,n}^r$ and show that the counterexamples identified in $\mathcal{M}_{g,n}^r$ do not occur within this substack. Moreover, we prove that $\mathcal{U}_{g,n}^r$ cannot be strictly contained in any other substack of $\mathcal{M}_{g,n}^r$ that admit a separated good moduli space. Furthermore, we show that the inclusion $\mathcal{U}_{g,n}^r \subset \mathcal{M}_{g,n}^r$ is both $\Theta$- and $\textsf{S}$-complete. These results will be used in a forthcoming paper to prove that $\mathcal{U}_{g,n}^r$ admits a separated, and indeed proper, good moduli space for $r \leq 5$.

math.AG↗
arXiv2026

On the fibers and semi-algebraicity of ReLU neuromanifolds

We study the semi-algebraicity of the neuromanifold $\mathcal{M}_\mathbf{d}$ of a feedforward ReLU neural network and its symmetries. We prove that $\mathcal{M}_\mathbf{d}$ is not a semi-algebraic quotient of the space of weights of the network. We introduce and study the notion of \emph{honest} open subset of the space of weights, where the network does not show any hidden symmetries. Finally, we conjecture that the maximal honest open is always semi-algebraic and prove that in the shallow case it is even Zariski.

math.AG↗
arXiv2026

Good Moduli Spaces in Derived Algebraic Geometry

We develop a theory of good moduli spaces for derived Artin stacks, which naturally generalizes the classical theory of good moduli spaces introduced by Alper. As such, many of the fundamental results and properties regarding good moduli spaces for classical Artin stacks carry over to the derived context. In fact, under natural assumptions, often satisfied in practice, we show that the derived theory essentially reduces to the classical theory. As applications, we establish derived versions of the étale slice theorem for good moduli spaces and the partial desingularization procedure of good moduli spaces.

math.AG↗
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.

math.AG↗
arXiv2026

The local geometry of the stack of $A_r$-stable curves

In this paper we study the local geometry of the stack of pointed $A_r$-stable curves. In particular, we analyze the deformation theory of $A_r$-stable curves and their automorphism groups in order to study the combinatorics of families of curves over $[\mathbb{A}^1/\mathbb{G}_m]$, and use this to classify all closed points of the stack of $A_r$-stable curves. As a byproduct, we also classify all open substacks of the moduli stack of degree $2$ cyclic covers of $\mathbb{P}^1$ that admit a separated good moduli space. This is the first in a series of three papers aimed at studying obstructions for the existence of good moduli spaces for stacks of curves with $A$-type singularities, and using these to find an open substack of the stack of $A_r$-stable curves that admits a proper non-projective good moduli space when $r=5$.

math.AG↗
arXiv2023

The derived Brauer map via twisted sheaves

Let $X$ be a quasicompact quasiseparated scheme. The collection of derived Azumaya algebras in the sense of Toën forms a group, which contains the classical Brauer group of $X$ and which we call $Br^\dagger(X)$ following Lurie. Toën introduced a map $ϕ:Br^\dagger(X)\to H^2_{et}(X,\mathbb G_m)$ which extends the classical Brauer map, but instead of being injective, it is surjective. In this paper we study the restriction of $ϕ$ to a subgroup $Br(X)\subset Br^\dagger(X)$, which we call the "derived Brauer group", on which $ϕ$ becomes an isomorphism $Br(X)\simeq H^2_{et}(X,\mathbb G_m)$. This map may be interpreted as a derived version of the classical Brauer map which offers a way to "fill the gap" between the classical Brauer group and the cohomogical Brauer group. The group $Br(X)$ was introduced by Lurie by making use of the theory of prestable $\infty$-categories. There, the mentioned isomorphism of abelian groups was deduced from an equivalence of $\infty$-categories between the "Brauer space" of invertible presentable prestable $\mathcal O_X$-linear categories, and the space $Map(X,K(\mathbb G_m,2))$. We offer an alternative proof of this equivalence of $\infty$-categories, characterizing the functor from the left to the right via gerbes of connective trivializations, and its inverse via connective twisted sheaves. We also prove that this equivalence carries a symmetric monoidal structure, thus proving a conjecture of Binda an Porta.

math.AG↗
arXiv2023

The (almost) integral Chow ring of $\overline{\mathcal{M}}_3$

This paper is the fourth in a series of four papers aiming to describe the (almost integral) Chow ring of $\overline{\mathcal{M}}_3$, the moduli stack of stable curves of genus $3$. In this paper, we finally compute the Chow ring of $\overline{\mathcal{M}}_3$ with $\mathbb{Z}[1/6]$-coefficients.

math.AG↗
arXiv2023

The (almost) integral Chow ring of $\widetilde{\mathcal{M}}_3^7$

This paper is the third in a series of four papers aiming to describe the (almost integral) Chow ring of $\overline{\mathcal{M}}_3$, the moduli stack of stable curves of genus $3$. In this paper, we compute the Chow ring of $\widetilde{\mathcal{M}}_3^7$ with $\mathbb{Z}[1/6]$-coefficients.

math.AG↗
arXiv2023

Hyperelliptic $A_r$-stable curves (and their moduli stack)

This paper is the second in a series of four papers aiming to describe the (almost integral) Chow ring of $\Mbar_3$, the moduli stack of stable curves of genus $3$. In this paper, we introduce the moduli stack $\Htilde_g^r$ of hyperelliptic $A_r$-stable curves and generalize the theory of hyperelliptic stable curves to hyperelliptic $A_r$-stable curves. In particular, we prove that $\Htilde_g^r$ is a smooth algebraic stacks which can be described using cyclic covers of twisted curves of genus $0$ and it embeds in $\Mtilde_g^r$ (the moduli stack of $A_r$-stable curves) as the closure of the moduli stack of smooth hyperelliptic curves.

math.AG↗
arXiv2023

The moduli stack of $A_r$-stable curves

This paper is the first in a series of four papers aiming to describe the (almost integral) Chow ring of $\bar{\mathcal{M}}_3$, the moduli stack of stable curves of genus $3$. In this paper, we introduce the moduli stack $\tilde{\mathcal{M}}_{g,n}^r$ of $n$-pointed $A_r$-stable curves and extend some classical results about $\bar{\mathcal{M}}_{g,n}$ to $\tilde{\mathcal{M}}_{g,n}^r$, namely the existence of the contraction morphism. Moreover, we describe the normalization of the locally closed substack of $\tilde{\mathcal{M}}_{g,n}^r$ parametrizing curves with $A_h$-singularities for a fixed $h\leq r$.

math.AG↗
arXiv2023

$A_r$-stable curves and the Chow ring of $\overline{\mathcal{M}}_3$

In this work, we introduce the moduli stack $\widetilde{\mathcal{M}}_{g,n}^r$ of $n$-pointed, $A_r$-stable curves of genus $g$ and use it to compute the Chow ring of $\overline{\mathcal{M}}_3$. As a byproduct, we also compute the Chow ring of $\widetilde{\mathcal{M}}_3^7$. All the Chow rings are assumed to be with coefficients in $\mathbb{Z}[1/6]$.

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