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Sebastian Bozlee

Publications and source records attributed to Sebastian Bozlee.

9 recordsLinked to original sources
arXiv2026

Close connectedness of the moduli stack of reduced curves

We prove that the moduli stack of all reduced $n$-pointed curves is ``closely connected'' in characteristic zero, in the sense that each irreducible component of the stack intersects the component of smoothable curves. We achieve this by performing a detailed study of Ishii's territories, moduli schemes parametrizing reduced curve singularities together with a normalization map. We give explicit equations for territories, bound their dimensions, describe certain functoriality properties, and study the action of several groups on territories. Along the way, we prove the existence of non-smoothable reduced curve singularities in new ranges, generalizing work of Mumford, Pinkham, Greuel, and Stevens.

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arXiv2023

A stratification of moduli of arbitrarily singular curves

We introduce a new moduli stack $\mathscr{E}_{g,n}$ of "equinormalized curves," closely related to the moduli space of all reduced, connected algebraic curves. We construct a stratification $\bigsqcup_\Gamma \mathscr{E}_\Gamma$ of $\mathscr{E}_{g,n}$ indexed by generalized dual graphs and prove that each stratum $\mathscr{E}_{\Gamma}$ is a fiber bundle over a finite quotient of a product of $\moduli_{g,n}$s. The fibers are moduli schemes parametrizing subalgebras of a fixed algebra, and are in principle explicitly computable as locally closed subschemes of products of Grassmannians. We thus obtain a remarkably explicit geometric description of moduli of reduced curves with arbitrary singularities.

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arXiv2023

Hyperelliptic Gorenstein curves and logarithmic differentials

We produce a flexible tool for contracting subcurves of logarithmic hyperelliptic curves, which is local around the subcurve and commutes with arbitrary base-change. As an application, we prove that hyperelliptic multiscale differentials determine a sequence of Gorenstein contractions of the underlying nodal curve, whose dualising bundle they descend to generate. This is the first piece of evidence for a more general conjecture about limits of differentials.

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arXiv2022

On compactifications of $\mathcal{M}_{g,n}$ with colliding markings

In this paper, we study all ways of constructing modular compactifications of the moduli space $\mathcal{M}_{g,n}$ of $n$-pointed smooth algebraic curves of genus $g$ by allowing markings to collide. We find that for any such compactification, collisions of markings are controlled by a simplicial complex which we call the collision complex. Conversely, we identify modular compactifications of $\mathcal{M}_{g,n}$ with essentially arbitrary collision complexes, including complexes not associated to any space of weighted pointed stable curves. These moduli spaces classify the modular compactifications of $\mathcal{M}_{g,n}$ by nodal curves with smooth markings as well as the modular compactifications of $\mathcal{M}_{1,n}$ with Gorenstein curves and smooth markings. These compactifications generalize previous constructions given by Hassett, Smyth, and Bozlee--Kuo--Neff.

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arXiv2022

The Scheme of Monogenic Generators II: Local Monogenicity and Twists

This is the sequel paper to arXiv:2108.07185, continuing a study of monogenicity of number rings from a moduli-theoretic perspective. By the results of the first paper in this series, a choice of a generator $\theta$ for an $A$-algebra $B$ is a point of the scheme $\mathcal{M}_{B/A}$. In this paper, we study and relate several notions of local monogenicity that emerge from this perspective. We first consider the conditions under which the extension $B/A$ admits monogenerators locally in the Zariski and finer topologies, recovering a theorem of Pleasants as a special case. We next consider the case in which $B/A$ is \'etale, where the local structure of \'etale maps allows us to construct a universal monogenicity space and relate it to an unordered configuration space. Finally, we consider when $B/A$ admits local monogenerators that differ only by the action of some group (usually $\mathbb{G}_m$ or $\mathrm{Aff}^1$), giving rise to a notion of twisted monogenerators. In particular, we show a number ring $A$ has class number one if and only if each twisted monogenerator is in fact a global monogenerator $\theta$.

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arXiv2021

The Scheme of Monogenic Generators I: Representability

This is the first in a series of two papers that study monogenicity of number rings from a moduli-theoretic perspective. Given an extension of algebras $B/A$, when is $B$ generated by a single element $\theta \in B$ over $A$? In this paper, we show there is a scheme $\mathcal{M}_{B/A}$ parameterizing the choice of a generator $\theta \in B$, a "moduli space" of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples.

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arXiv2021

A classification of modular compactifications of the space of pointed elliptic curves by Gorenstein curves

We classify the Deligne-Mumford stacks M compactifying the moduli space of smooth $n$-pointed curves of genus one under the condition that the points of M represent Gorenstein curves with distinct markings. This classification uncovers new moduli spaces $\bar{\mathcal{M}}_{1,n}(Q)$, which we may think of coming from an enrichment of the notion of level used to define Smyth's $m$-stable spaces. Finally, we construct a cube complex of Artin stacks interpolating between the $\bar{\mathcal{M}}_{1,n}(Q)$'s, a multidimensional analogue of the wall-and-chamber structure seen in the log minimal model program for $\bar{\mathcal{M}}_g$.

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arXiv2021

Contractions of subcurves of families of log curves

Let $C$ be a nodal curve, and let $E$ be a union of semistable subcurves of $C$. We consider the problem of contracting the connected components of $E$ to singularities in a way that preserves the genus of $C$ and makes sense in families, so that this contraction may induce maps between moduli spaces of curves. In order to do this, we introduce the notion of mesa curve, a nodal curve $C$ with a logarithmic structure and a piecewise linear function $\overlineλ$ on the tropicalization of $C$. This piecewise linear function determines a subcurve $E$. We then construct a contraction of $E$ inside of $C$ for families of mesa curves. Resulting singularities include the elliptic Gorenstein singularities.

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