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Teppei Takamatsu

Publications and source records attributed to Teppei Takamatsu.

At least 19 recordsLinked to original sources
arXiv2026

Some generalizations of Oort's conjecture

For a prime $p\geq 5$, let $\mathscr S_g$ be the moduli space over $\overline{\mathbb F}_p$ of $g$-dimensional principally polarized supersingular abelian varieties. We show that each of the following loci contains an open dense subscheme on which the principally polarized abelian varieties have automorphism group $\{\pm1\}$: (i) certain supersingular Ekedahl--Oort strata when $g$ is even, (ii) the loci in $\mathscr S_g$ with non-supersingular Ekedahl--Oort invariants of positive Coxeter type when $g\geq 3$, and (iii) the locus in $\mathscr S_g$ with $a$-number at least $2$ when $g\geq 4$. Consequently, for $g\geq 4$, the complement in $\mathscr S_g$ of the open locus where the automorphism group is $\{\pm1\}$ has codimension at least $2$. These results confirm Oort's conjecture for $p\geq 5$. We reduce them to statements about affine Deligne--Lusztig varieties for $\operatorname{GSp}_{2g}$ and prove analogues of (ii) for $\operatorname{GL}_{2g}$ and $\operatorname{GSO}_{4m}$.

math.AG↗
arXiv2026

Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture

We investigate in this paper the so-called pointed Shafarevich problem for families of primitive symplectic varieties. More precisely, for any fixed pointed curve $(B, 0)$ and any fixed primitive symplectic variety $X$, among all locally trivial families of $\mathbb{Q}$-factorial and terminal primitive symplectic varieties over $B$ whose fiber over $0$ is isomorphic to $X$, we show that there are only finitely many isomorphism classes of generic fibers. Moreover, assuming semi-ampleness of isotropic nef divisors, which holds true for all hyper-Kähler manifolds of known deformation types, we show that there are only finitely many such projective families up to isomorphism. These results are optimal since we can construct infinitely many pairwise non-isomorphic (not necessarily projective) families of smooth hyper-Kähler varieties over some pointed curve $(B, 0)$ such that they are all isomorphic over the punctured curve $B\backslash \{0\}$ and have isomorphic fibers over the base point $0$.

math.AG↗
arXiv2026

Perfectoid pure thresholds of lifts of rational double points

We study the perfectoid pure threshold with respect to $p$, an invariant of singularities in mixed characteristic $(0,p)$ arising from perfectoid purity. In this paper, we compute perfectoid pure thresholds for lifts of rational double points. We show that the set of such thresholds is contained in $\mathbb{Q}$ and satisfies the ascending chain condition. In characteristic $2$, all reciprocals of positive integers occur, and $0$ is the unique accumulation point.

math.AG↗
arXiv2026

Fano threefolds of genus 12 with large automorphism group in positive and mixed characteristic

We study prime Fano threefolds of genus 12 ($V_{22}$-varieties) with positive-dimensional automorphism groups in positive and mixed characteristic. We classify such varieties over any perfect field. In particular, we prove that $V_{22}$-varieties of Mukai-Umemura type over $k$ exist if and only if $\mathrm{char}\ k \neq 2$, $5$. We also prove the same result for $\mathbb{G}_a$-type. As arithmetic applications, we show that the Shafarevich conjecture holds for $V_{22}$-varieties of Mukai-Umemura type and of $\mathbb{G}_m$-type, while it fails for $V_{22}$-varieties of $\mathbb{G}_a$-type. Moreover, we prove that there exists $V_{22}$-varieties over $\mathbb{Z}$, whereas there do not exist $V_{22}$-varieties over $\mathbb{Z}$ whose generic fiber has a positive-dimensional automorphism group.

math.AG↗
arXiv2026

Quasi-$F^{\infty}$-split height versus quasi-$F$-regular height for rational double points and graded rings

In this paper, we study a phenomenon concerning quasi-$F$-singularities: under suitable hypotheses, the finiteness of the quasi-$F^{\infty}$-split height ($\mathrm{ht}^{\infty}$) implies quasi-$F$-regularity, and moreover, $\mathrm{ht}^{\infty}$ coincides with the quasi-$F$-regular height ($\mathrm{ht}^{\mathrm{reg}}$). We establish this coincidence for two important classes of isolated Gorenstein singularities. First, we explicitly compute $\mathrm{ht}^{\infty}$ and $\mathrm{ht}^{\mathrm{reg}}$ for all rational double points, showing that every non-$F$-pure rational double point satisfies $\mathrm{ht}^\infty = \mathrm{ht}^{\mathrm{reg}}$. Second, for localizations of graded non-$F$-pure normal Gorenstein rings with $F$-rational punctured spectrum, we again obtain the equality $\mathrm{ht}^\infty = \mathrm{ht}^{\mathrm{reg}}$.

math.AC↗
arXiv2025

On the semi-infinite Deligne--Lusztig varieties for $\mathrm{GSp}$

We prove that Lusztig's semi-infinite Deligne--Lusztig variety for $\mathrm{GSp}$ (and its inner form) is isomorphic, as a set with action, to an affine Deligne--Lusztig variety at infinite level, generalizing a result of Chan--Ivanov. Furthermore, we show that a component of some affine Deligne--Lusztig variety $X^0_{w_r}(b)_{\mathcal{L}}$ for $\mathrm{GSp}$ can be written, up to perfection, as a direct product of a classical Deligne--Lusztig variety with an affine space. We also study the varieties $X_h$ defined by Chan and Ivanov, and show that $X_h$ at infinite level can be realized as a subset of semi-infinite Deligne--Lusztig varieties defined using components of affine Deligne--Lusztig varieties such as $X^0_{w_r}(b)_{\mathcal{L}}$ above, even in the $\mathrm{GSp}$ case. This reinterprets previous constructions of representations from $X_h$ as instances of Lusztig's conjectural picture.

math.AG↗
arXiv2025

Fedder type criteria for quasi-$F$-splitting I

Yobuko recently introduced the notion of quasi-$F$-splitting and quasi-$F$-split heights, which generalize and quantify the notion of Frobenius-splitting, and proved that quasi-$F$-split heights coincide with Artin-Mazur heights for Calabi-Yau varieties. In this paper, we prove Fedder type criteria for quasi-$F$-splittings of complete intersections, and in particular, obtain a simple formula to compute Artin-Mazur heights of Calabi-Yau hypersurfaces. As one of its applications, we prove that there exist Calabi-Yau varieties of arbitrarily high Artin-Mazur height over $\mathbb{F}_2$. We also give explicit defining equations of quartic K3 surfaces over $\mathbb{F}_{3}$ realizing all the possible Artin-Mazur heights.

math.AG↗
arXiv2025

Fedder type criteria for quasi-$F$-splitting II

In this paper, we apply Fedder-type criteria for quasi-$F$-splitting to provide explicit computations of quasi-$F$-split heights for Calabi-Yau hypersurfaces, bielliptic surfaces, Fano varieties, and rational double points. We also find interesting phenomena concerned with inversion of adjunction, fiber products, Fano varieties, and general fibers of fibrations.

math.AG↗
arXiv2025

Quintic del Pezzo threefolds in positive and mixed characteristic

We show that smooth quintic del Pezzo threefolds over arbitrary base schemes are classified by non-degenerate ternary symmetric bilinear forms. Then we describe the automorphism group schemes, the Hilbert schemes of lines and the orbit structures of quintic del Pezzo threefolds, and we find several new phenomena in characteristic two. As arithmetic applications, we prove a refinement of the Shafarevich conjecture, and prove that there are exactly two isomorphism classes of quintic del Pezzo threefolds over the ring of rational integers.

math.AG↗
arXiv2025

Arithmetic finiteness of Mukai varieties of genus 7

We study arithmetic finiteness of prime Fano threefolds of genus 7 and their higher dimensional generalization, called Mukai varieties of genus 7. For prime Fano threefolds of genus 7, we provide an arithmetic refinement of the Torelli theorem, obtain Shafarevich-type finiteness results, and show the failure of the Néron--Ogg--Shafarevich criterion of good reduction. For Mukai varieties of genus 7, we prove that Shafarevich-type finiteness results hold in dimensions 9 and 10, but fail in dimension 6. In addition, we show that Mukai $n$-folds of genus 7 over $\mathbb{Z}$ do not exist for $n \leq 4$, whereas they exist for $5 \leq n \leq 10$.

math.AG↗
arXiv2025

Arithmetic monodromy of hyper-Kähler varieties over $p$-adic fields

In this paper, we study the $p$-adic and $\ell$-adic monodromy operators associated with hyper-Kähler varieties over $p$-adic fields, in connection with Looijenga-Lunts-Verbitsky Lie algebras. We investigate a conjectural relation between the nilpotency indices of these monodromy operators on higher-degree cohomology groups and on the second cohomology, which may be viewed as an arithmetic analogue of Nagai's conjecture for degenerations of hyper-Kähler manifolds over a disk. We verify this arithmetic version of Nagai's conjecture for hyper-Kähler varieties over $p$-adic fields, assuming they belong to one of the four known deformation types. As part of our approach, we introduce a new method to analyze the $p$-adic cohomology of hyper-Kähler varieties via Sen's theory.

math.AG↗
arXiv2025

Reduction of bielliptic surfaces

A bielliptic surface (or hyperelliptic surface) is a smooth surface with a numerically trivial canonical divisor such that the Albanese morphism is an elliptic fibration. In the first part of this paper, we study the structure of bielliptic surfaces over a field of characteristic different from $2$ and $3$, in order to prove the Shafarevich conjecture for bielliptic surfaces with rational points. Furthermore, we demonstrate that the Shafarevich conjecture generally fails for bielliptic surfaces without rational points. In particular, this paper completes the study of the Shafarevich conjecture for minimal surfaces of Kodaira dimension $0$. In the second part of this paper, we study a Néron model of a bielliptic surface. We establish the potential existence of a Néron model for a bielliptic surface when the residual characteristic is not equal to $2$ or $3$.

math.AG↗