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Quentin Posva

Publications and source records attributed to Quentin Posva.

12 recordsLinked to original sources
arXiv2026

Local stability of Frobenius and Artin--Schreier base-change: a comparison

Let $(X,\Delta)\to C$ be a family of pairs over a smooth curve in positive characteristic $p>0$. We study the discrepancies of divisors over the base-change $(X,\Delta)\to C$ along Artin--Schreier and Frobenius covers of $C$, through ramification data for Artin--Schreier and degree $p$ purely inseparable extensions of DVRs. As an application, we recover a result of Hu and Zong about permanence of local stability for such base-change.

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arXiv2026

Base-change of locally stable families in positive characteristic

We investigate the permanence of local stability for one-parameter families $X\to C$ under finite flat base-changes, when the base-field has positive characteristic $p>0$. Building on previous work of Hu--Zong, we show that it suffices to consider base-changes by Frobenius morphisms. In that case, we show that the situation is governed by the discrepancies of the pairs $(X,X_c)$, together with some differential invariants of the vertical divisors whose multiplicity in their fiber is divisible by $p$. While the behaviour of these invariants remains in general mysterious, we establish upper bounds under some $F$-splitting assumptions.

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arXiv2026

On linear $\alpha_p$-quotients

We study linear $\alpha_p$-actions on affine spaces and the associated quotient singularities, using explicit stacky resolutions. We describe when the quotient singularities are log canonical, canonical or terminal, and we compute their stringy motivic invariants. The second author and Fabio Tonini conjectured that these invariants coincide with those of linear $\mathbb{Z}/p$-quotients: our approach reduces this conjecture to an equality of explicit multi-sets, which we check for a large number of primes using a computer software. A general proof of the equality of multi-sets is given in the appendix written by Linus R\"osler.

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arXiv2025

Pathological MMP singularities as $\alpha_p$-quotients

We construct pathological examples of MMP singularities in every positive characteristic using quotients by $\alpha_p$-actions. In particular, we obtain non-$S_3$ terminal singularities, as well as locally stable (respectively stable) families whose general fibers are smooth (respectively klt, Cohen--Macaulay and $F$-injective) and whose special fibers are non-$S_2$. The dimensions of these examples are bounded below by a linear function of the characteristic.

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arXiv2024

Resolution of $1$-foliations singularities on surfaces and threefolds

We consider resolution of singularities for $1$-foliations on varieties of dimension at most three in positive characteristic. We prove that such singularities can be completely resolved if we allow tame regular Deligne--Mumford stacks as underlying spaces. If one restricts to underlying varieties, we show that $1$-foliations singularities can be simplified into multiplicative ones.

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arXiv2023

On the singularities of quotients by 1-foliations

We study the singularities of varieties obtained as infinitesimal quotients by $1$-foliations in positive characteristic. (1) We show that quotients by (log) canonical $1$-foliations preserve the (log) singularities of the MMP. (2) We prove that quotients by multiplicative derivations preserve many properties, amongst which most $F$-singularities. (3) We formulate a notion of families of $1$-foliations, and investigate the corresponding families of quotients.

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arXiv2023

Normality of minimal log canonical centers of threefolds in mixed and positive characteristic

We prove the normality of minimal log canonical centers on threefold pairs which residue fields are perfect of residue characteristics $p\neq 2,3 $ and $5$. We also show that the union of all log canonical centers on threefold pairs with standard coefficients are seminormal provided that the residue characteristic is large enough. We provide an example of a non-seminormal log canonical center on a threefold in characteristic $3$, and give sufficient conditions to construct similar examples.

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arXiv2021

Abundance for slc surfaces over arbitrary fields

We prove the abundance conjecture for projective slc surfaces over arbitrary fields of positive characteristic. The proof relies on abundance for lc surfaces over abritrary fields, proved by Tanaka, and on the technique of Hacon and Xu to descend semi-ampleness from the normalization. We also present applications to dlt threefold pairs, and to mixed characteristic families of surfaces.

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arXiv2021

Gluing for stable families of surfaces in mixed characteristic

We study the normalizations of non-normal stable families of slc surfaces over an excellent DVR. In mixed characteristic, we establish a gluing statement that is relevant for the properness of the moduli space of such surfaces. We also study the fibers of such glued families, in mixed and equi-characteristic, and provide an essential result for slc adjunction in residue characteristic >5.

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arXiv2021

Gluing theory for slc surfaces and threefolds in positive characteristic

We develop a gluing theory in the sense of Koll\'{a}r for slc surfaces and threefolds in positive characteristic. For surfaces we are able to deal with every positive characteristic $p$, while for threefolds we assume that $p>5$. Along the way we study nodes in characteristic $2$ and establish a theory of sources and springs \`a la Koll\'{a}r for threefolds. We also give applications to the topology of lc centers on slc threefolds, and to the projectivity of the moduli space of stable surfaces in characteristic $p>5$.

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arXiv2019

Positivity of the CM line bundle for K-stable log Fanos

We prove the bigness of the Chow-Mumford line bundle associated to a $\mathbb{Q}$-Gorenstein family of log Fano varieties of maximal variation with uniformly K-stable general geometric fibers. This result generalizes a recent theorem of Codogni and Patakfalvi to the logarithmic setting.

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