arXiv subjects

Tomoki Oda

Publications and source records attributed to Tomoki Oda.

3 recordsLinked to original sources

arXiv2026

Computing Cox rings via the cone conjecture

We initiate a program to study the Cox ring of Calabi-Yau varieties, employing the notion of Morrison-Kawamata dream spaces. In this setting, we establish an analogue of the Hu-Keel GIT constructions for Mori dream spaces. More precisely, for a Morrison-Kawamata dream space $X$, we establish a correspondence between the small $\mathbb{Q}$-factorial modifications of $X$ and the GIT quotients of $\operatorname{Spec}\operatorname{Cox}(X)$. We further show that the Cox ring of a Morrison-Kawamata dream space is a filtered direct limit of subalgebras, each of which is an inverse limit of finitely generated $\mathrm{Cl}(X)$-graded $\mathbb{K}$-algebras. As an application, we give an explicit presentation of the Cox ring of a very general hypersurface of multidegree $(2,\dots,2,n+1)$ in $(\mathbb{P}^1)^m\times \mathbb{P}^n$. Furthermore, we prove that the Cox ring of such a hypersurface is of dense $F$-pure type.

math.AG↗

arXiv2026

Bianchi groups and automorphisms of rank-four $K3$ surfaces

We relate the arithmetic of Bianchi groups to automorphism groups of Picard-rank-four $K3$ surfaces. Let $K$ be an imaginary quadratic field with ring of integers $\mathcal O_K$, and let $S_K=\operatorname{Herm}_2(\mathcal O_K)$ be the rank-four lattice of $2\times2$ Hermitian matrices over $\mathcal O_K$, equipped with the quadratic form $2\det$. For an odd integer $N\geq1$, we consider a very general $S_K(2N)$-polarized $K3$ surface $X_{K,2N}$. We prove that its automorphism group is commensurable with a level-$2N$ congruence subgroup of the Bianchi group. Furthermore, we also obtain exact realizations of congruence subgroups as full automorphism groups. Namely, if $K=\mathbb Q(i)$ or $K=\mathbb Q(\sqrt{-p})$, where $p$ is prime, then \[ \operatorname{Aut}(X_{K,2}) \cong P\Gamma_K(2). \] Thus, for every prime $p$, the projective principal congruence subgroup of level $2$ over $\mathcal O_{\mathbb Q(\sqrt{-p})}$ occurs as the full automorphism group of a Picard-rank-four $K3$ surface. At higher levels, the full automorphism group may be either the projective principal congruence subgroup or the strictly larger projective level subgroup $\operatorname{Bi}_K(2N)$, depending on the arithmetic of the primes dividing the level. We further explain these arithmetic groups geometrically. The surfaces $X_{K,2}$ arise as deformations of the Kummer surfaces $\operatorname{Km}(E_K\times E_K)$, yielding explicit double-cover models and genus-one fibrations. For $K=\mathbb Q(\sqrt{-2})$ and $K=\mathbb Q(\sqrt{-7})$, the automorphism group is generated by Mordell--Weil translations associated with genus-one fibrations coming from cusps, together with the covering involution. For $K=\mathbb Q(i)$ and $K=\mathbb Q(\sqrt{-3})$, we construct complete-intersection models in products of projective spaces and show that their automorphism groups are generated by deck involutions.

math.AG↗

arXiv2025

Geometry of tropical mutation surfaces with a single mutation

Escobar, Harada, and Manon introduced polyptych lattices as a piecewise-linear extension of the lattice-polytope formalism of toric geometry. In this paper we study the first genuinely non-toric case: rank-two polyptych lattices with a single shear. A detropicalization is given by a polynomial $f(y)$, and the corresponding affine surface is $U_f=\operatorname{Spec} K[x_1,x_2,y^{\pm 1}]/\langle x_1x_2-f(y)\rangle.$ We classify these detropicalizations, compute the complexity of their projective compactifications, and show that the resulting log Calabi--Yau surface pairs are of cluster type. Conversely, we prove that every normal projective $\mathbb Q$-factorial index-one log Calabi--Yau surface pair with reduced boundary, ample boundary support, and a nontrivial $\mathbb G_m$-action arises from this single-shear construction. We also construct a global family interpolating between the two toric degenerations associated with the two charts, and compute the Cox rings of the resulting tropical mutation surfaces.

math.AG↗