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Akihiro Kanemitsu

Publications and source records attributed to Akihiro Kanemitsu.

15 recordsLinked to original sources
arXiv2026

Prime Fano threefolds of genus 8 in positive characteristic

We prove that a prime Fano threefold of genus 8 over an algebraically closed field of positive characteristic is isomorphic to a linear section of the Grassmannian variety Gr(2, 6). As applications, it is shown that a prime Fano threefold of genus 8 is irrational and, if the characteristic is larger than two, then it is globally F-regular.

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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.

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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.

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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 $5 \leq n \leq 10$.

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arXiv2023

Polarized K3 surfaces of genus thirteen and curves of genus three

We describe a general (primitively) polarized K3 surface $(S,h)$ with $(h^2)=24$ as a complete intersection variety with respect to vector bundles on the $6$-dimensional moduli space $\mathcal{N}^-$ of the stable vector bundles of rank two with fixed odd determinant on a curve $C$ of genus $3$. If the curve $C$ is hyperelliptic, then $\mathcal{N}^-$ is a subvariety of the $12$-dimensional Grassmann variety $\operatorname{Gr}(\mathbf{C}^8,2)$ defined by a pencil of quadric forms. In this case, our description implies that a general $(S,h)$ is the intersection of two (7-dimensional) contact homogeneous varieties of $\operatorname{Spin}(7)$ in the Grassmann variety $\operatorname{Gr}(\mathbf{C}^8,2)$.

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arXiv2020

Projective varieties with nef tangent bundle in positive characteristic

Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that $X$ admits a smooth morphism $X \to M$ such that the fibers are Fano varieties with nef tangent bundle and $T_M$ is numerically flat. We also prove that extremal contractions exist as smooth morphisms. As an application, we prove that, if the Frobenius morphism can be lifted modulo $p^2$, then $X$ admits, up to a finite étale Galois cover, a smooth morphism onto an ordinary abelian variety whose fibers are products of projective spaces.

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arXiv2019

Fano manifolds and stability of tangent bundles

We determine the stability/instability of the tangent bundles of the Fano varieties in a certain class of two orbit varieties, which are classified by Pasquier in 2009. As a consequence, we show that some of these varieties admit unstable tangent bundles, which disproves a conjecture on stability of tangent bundles of Fano manifolds.

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arXiv2018

Mukai pairs and simple $K$-equivalence

A $K$-equivalent map between two smooth projective varieties is called simple if the map is resolved in both sides by single smooth blow-ups. In this paper, we will provide a structure theorem of simple $K$-equivalent maps, which reduces the study of such maps to that of special Fano manifolds. As applications of the structure theorem, we provide examples of simple $K$-equivalent maps, and classify such maps in several cases, including the case of dimension at most $8$.

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arXiv2018

Classification of Mukai pairs with dimension $4$ and rank $2$

We give the complete classification of Mukai pairs of dimension $4$ and rank $2$ with Picard number one, that is, pairs $(X,E)$ where $X$ is a Fano $4$-fold with Picard number one, and $E$ is an ample vector bundle of rank two on $X$ with $c_1(X) = c_1(E)$. Equivalently, the present paper completes the classification of ruled Fano $5$-folds with index two, which was partially done by C. Novelli and G. Occhetta in 2007.

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arXiv2017

Classification of Mukai pairs with corank $3$

We classify pairs $(X,\mathscr E)$ where $X$ is a smooth Fano manifold of dimension $n \geq 5$ and $\mathscr E$ is an ample vector bundle of rank $n-2$ on $X$ with $c_1(\mathscr E) = c_1(X)$.

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arXiv2016

Extremal rays and nefness of tangent bundles

In view of Mori theory, rational homogenous manifolds satisfy a recursive condition: every elementary contraction is a rational homogeneous fibration and the image of any elementary contraction also satisfies the same property. In this paper, we show that a smooth Fano $n$-fold with the same condition and Picard number greater than $n-6$ is either a rational homogeneous manifold or the product of $n-7$ copies of $\mathbb{P}^1$ and a Fano $7$-fold $X_0$ constructed by G. Ottaviani. We also clarify that $X_0$ has non-nef tangent bundle and in particular is not rational homogeneous.

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