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Hiromichi Takagi

Publications and source records attributed to Hiromichi Takagi.

At least 19 recordsLinked to original sources
arXiv2026

Key variety construction of Sarkisov links for prime $\mathbb{Q}$-Fano threefolds of codimension four associated to Type ${\rm II}_{2}$ projections

In our paper [Tak6], we constructed eight families of quasi-smooth prime $\mathbb{Q}$-Fano threefolds, anticanonically embedded in codimension four, using weighted projectivizations of the $14$-dimensional affine variety $Π_{\mathbb{A}}^{14}$or its cone. Let $\widehat{f}\colon\widehat{X}\to X$ be the unique divisorial extraction at one specified singularity of maximal index. In this paper, we explicitly construct the Sarkisov link starting from $\widehat{f}$ for $X$ belonging to seven of these families. This is achieved by using the Sarkisov link associated with the weighted projectivization of $Π_{\mathbb{A}}^{14}$ or its cone corresponding to $X$. As a consequence, we show that the Sarkisov link ends with either a fibration whose general fiber is a del Pezzo surface of degree one or a divisorial contraction of type $(2,1)$ to weighted complete intersections of codimension at most two. We also provide more detailed descriptions of these Sarkisov links.

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arXiv2025

Constructing and characterizing prime $\mathbb{Q}$-Fano threefolds of genus one and with six $1/2(1,1,1)$-singularites via key varieties

We consider the classification problem of prime $\mathbb{Q}$-Fano 3-folds with at most $1/2(1,1,1)$-singularities, which was initiated in [Taka2]. We construct two distinct classes of such 3-folds with genus one and six $1/2(1,1,1)$-singularities, each equipped with a prescribed Sarkisov link. Our method involves constructing certain higher-dimensional $\mathbb{Q}$-Fano varieties $Σ$, referred to as key varieties, by extending the Sarkisov links to higher dimensions. We prove that each such 3-fold $X$ arises as a linear section of the corresponding key variety $Σ$, and conversely, any general linear section of $Σ$ yields such an $X$. Various geometric properties of the key varieties $Σ$ are also investigated and clarified.

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arXiv2025

Key varieties for prime $\mathbb{Q}$-Fano threefolds defined by Freudenthal triple systems

In this paper, we concern with the classification of complex prime $\mathbb{Q}$-Fano $3$-folds of anti-canonical codimension 4 which are produced, as weighted complete intersections of appropriate weighted projectivizations of certain affine varieties related with $\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$-fibrations. Such affine varieties or their appropriate weighted projectivizations are called key varieties for prime $\mathbb{Q}$-Fano 3-folds. We realize that the equations of the key varieties can be described conceptually by Freudenthal triple systems (FTS, for short). The paper consists of two parts. In Part 1, we revisit the general theory of FTS; the main purpose of Part 1 is to derive the conditions of so called strictly regular elements in FTS so as to fit with our description of key varieties. Then, in Part 2, we define several key varieties for prime $\mathbb{Q}$-Fano 3-folds from the conditions of strictly regular elements in FTS. Among other things obtained in Part 2, we show that there exists a $14$-dimensional factorial affine variety $\mathfrak{U}_{\mathbb{A}}^{14}$ of codimension $4$ in an affine $18$-space with only Gorenstein terminal singularities, and we construct examples of prime $\mathbb{Q}$-Fano $3$-folds of No.20544 in [GRDB] as weighted complete intersections of the weighted projectivization of $\mathfrak{U}_{\mathbb{A}}^{14}$ in the weighted projective space $\mathbb{P}(1^{15},2^{2},3)$. We also clarify in Part 2 a relation between $\mathfrak{U}_{\mathbb{A}}^{14}$ and the $G_{2}^{(4)}$-cluster variety, which is a key variety for prime $\mathbb{Q}$-Fano 3-folds constructed in [CD].

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arXiv2025

Constructing prime $\mathbb{Q}$-Fano threefolds of codimension four via key varieties related with $\mathbb{P}^2\times \mathbb{P}^2$-fibrations

In our previous research, we constructed the affine varieties $Σ_{\mathbb{A}}^{13}$ and $Π_{\mathbb{A}}^{14}$ whose partial projectivizations admit $\mathbb{P}^{2}\times\mathbb{P}^{2}$-fibrations with relative Picard number one. In this paper, we produce prime quasi-smooth $\mathbb{Q}$-Fano 3-folds which are anticanonically embedded of codimension four and belong to 23 (resp.8) classes in the Graded Ring Database [GRDB], as weighted complete intersections in weighted projectivizations of $Σ_{\mathbb{A}}^{13}$ (resp.$Π_{\mathbb{A}}^{14}$ or its cone). We also show that a general member of the anticanonical linear system of a general prime $\mathbb{Q}$-Fano $3$-fold constructed in this way is a quasi-smooth $K3$ surface with at worst Du Val singularities.

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arXiv2025

Key varieties for prime $\mathbb{Q}$-Fano threefolds defined by Jordan algebras of cubic forms. Part I

We construct a $13$-dimensional affine variety $\mathscr{H}_{\mathbb{A}}^{13}$ associated with $\mathbb{P}^{2}\times\mathbb{P}^{2}$-fibrations of relative Picard number $1$. The construction is modelled on the fact that the affine cone over the Segre-embedded $\mathbb{P}^{2}\times\mathbb{P}^{2}$ is the null locus of the $\sharp$-map of the $9$-dimensional nondegenerate quadratic Jordan algebra $J$ of a cubic form. Using three fixed complementary primitive idempotents and the Peirce decomposition, we coordinatize $J$ by $8$ parameters and thereby obtain $\mathscr{H}_{\mathbb{A}}^{13}$. We then produce complex prime $\mathbb{Q}$-Fano $3$-folds, anticanonically embedded of codimension $4$, as weighted complete intersections in suitable weighted projectivizations of $\mathscr{H}_{\mathbb{A}}^{13}$, in its subvarieties, or in their weighted cones (allowing some coordinates of weight $0$). We refer to $\mathscr{H}_{\mathbb{A}}^{13}$ and these projectivizations as key varieties for prime $\mathbb{Q}$-Fano $3$-folds. As an application, we show that a prime $\mathbb{Q}$-Fano $3$-fold of genus $3$ with three $1/2(1,1,1)$-singularities of type No.,5.4 as in \cite{Tak1} arises as a linear section of a weighted projectivization of $\mathscr{H}_{\mathbb{A}}^{13}$ with all coordinates of positive weight, and conversely any such threefold is obtained in this way. Moreover, relating $\mathscr{H}_{\mathbb{A}}^{13}$ to the $C_{2}$-cluster variety of Coughlan--Ducat \cite{CD1}, we show that weighted projectivizations of $\mathscr{H}_{\mathbb{A}}^{13}$ or of its subvarieties serve as key varieties for prime $\mathbb{Q}$-Fano $3$-folds belonging to $108$ classes in the online database \cite{GRDB}.

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arXiv2024

Key varieties for prime $\mathbb{Q}$-Fano threefolds defined by Jordan algebras of cubic forms. Part II

Subsequent to the previous paper [Tak5], we are concerned with the classification of complex prime $\mathbb{Q}$-Fano $3$-folds of anti-canonical codimension 4 which are produced, as weighted complete intersections of appropriate weighted projectivizations of certain affine varieties related with $\mathbb{P}^{2}\times\mathbb{P}^{2}$-fibrations. Such affine varieties or their appropriate weighted projectivizations (possibly allowing some coordinates have weights $0$) are called key varieties for prime $\mathbb{Q}$-Fano 3-folds. The purpose of this paper is to give new constructions of a $14$-dimensional affine variety $Υ_{\mathbb{A}}^{14}$ and a $15$-dimensional affine variety $Π_{\mathbb{A}}^{15}$ related with $\mathbb{P}^{2}\times\mathbb{P}^{2}$-fibration, which were constructed and were shown to be key varieties in the papers [Tak3,Tak4] and [Tay]. It is well-known that the affine cone of the Segre embedded $\mathbb{P}^{2}\times\mathbb{P}^{2}$ is defined as the null loci of the so called $\sharp$-mapping of a 9-dimensional nondegenerate quadratic Jordan algebra $J$ of a cubic form. Inspired with this fact, we construct $Υ_{\mathbb{A}}^{14}$ and $Π_{\mathbb{A}}^{15}$ in the same way coordinatizing $J$ with 9 and 10 parameters, respectively. The coordinatization with $9$ parameters is derived by using a fixed primitive idempotent, and the associated Peirce decomposition. The coordinatization with $10$ parameters is derived from the construction of a quadratic Jordan subalgebra generated by $\sharp$-products of two elements due to Petersson [Pe1].

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arXiv2022

Duality Related with Key Varieties of $\mathbb{Q}$-Fano 3-folds. I

Abstract. In our previous paper arXiv:2210.16008, we show that any prime $\mathbb{Q}$-Fano 3-folds $X$ with only $1/2(1,1,1)$-singularities in certain 5 classes can be embedded as linear sections into bigger dimensional $\mathbb{Q}$-Fano varieties called key varieties, where each of the key varieties is constructed from certain data of the Sarkisov link staring from the blow-up at one $1/2(1,1,1)$-singularity of $X$. In this paper, we introduce varieties associated with the key varieties which are dual in a certain sense. As an application, we interpret a fundamental part of the Sarkisov link for each $X$ as a linear section of the dual variety. In a natural context describing the variety dual to the key variety of $X$ of genus 5 with one $1/2(1,1,1)$-singularity, we also characterize a general canonical curve of genus 9 with a $g_{7}^{2}$.

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arXiv2022

On Classification of $\mathbb{Q}$-Fano 3-folds of Gorenstein index 2. III

We classified prime $\mathbb{Q}$-Fano $3$-folds $X$ with only $1/2(1,1,1)$-singularities and with $h^{0}(-K_{X})\geq 4$ a long time ago. The classification was undertaken by blowing up each $X$ at one $1/2(1,1,1)$-singularity and constructing a Sarkisov link. The purpose of this paper is to reveal the geometries behind the Sarkisov links for $X$ in 5 classes. The main result asserts that any $X$ in the 5 classes can be embedded as linear sections into bigger dimensional $\mathbb{Q}$-Fano varieties called key varieties, where the key varieties are constructed by extending partially the Sarkisov link in higher dimensions.

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arXiv2022

Mirror Symmetry of Calabi-Yau Manifolds Fibered by (1,8)-Polarized Abelian Surfaces

We study mirror symmetry of a family of Calabi-Yau manifolds fibered by (1,8)-polarized abelian surfaces with Euler characteristic zero. By describing the parameter space globally, we find all expected boundary points (LCSLs), including those correspond to Fourier-Mukai partners. Applying mirror symmetry at each boundary point, we calculate Gromov-Witten invariants ($g\leq2$) and observe nice (quasi-)modular properties in their potential functions. We also describe degenerations of Calabi-Yau manifolds over each boundary point.

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arXiv2021

Key Varieties for Prime $\mathbb{Q}$-Fano Threefolds Related with $\mathbb{P}^{2}\times\mathbb{P}^{2}$-Fibrations. Part II

We construct a $15$-dimensional affine variety $Π_{\mathbb{A}}^{15}$ with a ${\rm GL}_{2}$- and $(\mathbb{C}^{*})^{4}$-actions. We denote by $Π_{\mathbb{A}}^{14}$ the affine variety obtained from $Π_{\mathbb{A}}^{15}$ by setting one specified variable to $1$ (we refer the precise definition to Definition 1.1 of the paper). Let $Π_{\mathbb{P}}^{13}$ be several weighted projectivizations of $Π_{\mathbb{A}}^{14}$, and $Π_{\mathbb{P}}^{14}$ the weighted cone over $Π_{\mathbb{P}}^{13}$ with a weight one coordinate added. We show that $Π_{\mathbb{P}}^{13}$ or $Π_{\mathbb{P}}^{14}$ produce, as weighted complete intersections, examples of prime $\mathbb{Q}$-Fano threefolds of codimension four belonging to the eight classes No.308, 501, 512, 550, 577, 872, 878, and 1766 of the graded ring database. The construction of $Π_{\mathbb{A}}^{15}$ is based on a certain type of unprojection and is inspired by R.Taylor's thesis submitted to University of Warwick. We also show that a partial projectivization of $Π_{\mathbb{A}}^{15}$ has a $\mathbb{P}^{2}\times\mathbb{P}^{2}$-fibration over the affine space $\mathbb{A}^{10}$. To show this, we introduce another $13$-dimensional affine variety $H_{\mathbb{A}}^{13}$ whose product with an open subset of $\mathbb{A}^{2}$ is isomorphic to a sextic cover of an open subset of $Π_{\mathbb{A}}^{15}$.

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arXiv2021

Key Varieties for Prime $\mathbb{Q}$-Fano Threefolds related with $\mathbb{P}^2\times \mathbb{P}^2$-Fibrations. Part I

We construct a $14$-dimensional affine variety $Σ^{14}_{\mathbb{A}}$ with a $\rm{GL}_3$- and a $(\mathbb{C}^*)^6$-actions. We denote by $Σ^{13}_{\mathbb{A}}$ the affine variety obtained from $Σ^{14}_{\mathbb{A}}$ by setting one specified variable to $1$ (we refer the precise definition to Definition 1.1 of the paper). We show that several weighted projectivizations of $Σ^{13}_{\mathbb{A}}$ and $Σ^{14}_{\mathbb{A}}$ produce, as weighted complete intersections, examples of prime $\mathbb{Q}$-Fano threefolds of codimension four belonging to $24$ classes of the graded ring database. Except No.360 in the database, these prime $\mathbb{Q}$-Fano threefolds have a Type I Tom projection. Moreover, they are not weighted complete intersections of the cluster variety of type $C_2$ introduced by Coughlan and Ducat. We also show that a partial projectivization of $Σ^{14}_{\mathbb{A}}$ has a $\mathbb{P}^2\times \mathbb{P}^2$-fibration over the affine space $\mathbb{A}^9$.

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arXiv2019

K3 surfaces from configurations of six lines in $\mathbb{P}^2$ and mirror symmetry I

From the viewpoint of mirror symmetry, we revisit the hypergeometric system $E(3,6)$ for a family of K3 surfaces. We construct a good resolution of the Baily-Borel-Satake compactification of its parameter space, which admits special boundary points (LCSLs) given by normal crossing divisors. We find local isomorphisms between the $E(3,6)$ systems and the associated GKZ systems defined locally on the parameter space and cover the entire parameter space. Parallel structures are conjectured in general for hypergeometric system $E(n,m)$ on Grassmannians. Local solutions and mirror symmetry will be described in a companion paper \cite{HLTYpartII}, where we introduce a K3 analogue of the elliptic lambda function in terms of genus two theta functions.

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arXiv2018

Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds

We study mirror symmetry of complete intersection Calabi-Yau manifolds which have birational automorphisms of infinite order. We observe that movable cones in birational geometry are transformed, under mirror symmetry, to the monodromy nilpotent cones which are naturally glued together.

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arXiv2015

Mirror Symmetry and Projective Geometry of Fourier-Mukai Partners

This is a survey article on mirror symmetry and Fourier-Mukai partners of Calabi-Yau threefolds with Picard number one based on recent works by the authors [HoTa1,2,3,4]. For completeness, mirror symmetry and Fourier-Mukai partners of K3 surfaces are also discussed.

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arXiv2015

Geometry of symmetric determinantal loci

We study algebro-geometric properties of determinantal loci of (n+1)th symmetric matrices and also their double covers for even ranks. Their singularities, Fano indices and birational geometries are studied in general. The double covers of symmetric determinantal loci of rank four are studied with special interest by noting their relation to the Hilbert schemes of conics on Grassmannians.

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arXiv2015

Towards Homological Projective duality for S^2 P^3 and S^2 P^4

We provide homological foundations to establish conjectural homological projective dualities between 1) S^2 P^3 and the double cover of the projective 9-space branched along the symmetric determinantal quartic, and 2) S^2 P^4 and the double cover of the symmetric determinantal quintic in the projective 14-space branched along the symmetric determinantal locus of rank at most 3.

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arXiv2015

Derived Categories of Artin-Mumford double solids

We consider the derived category of an Artin-Mumford quartic double solid blown-up at ten ordinary double points. We show that it has a semi-orthogonal decomposition containing the derived category of the Enriques surface of a Reye congruence. This answers affirmatively a conjecture by Ingalls and Kuznetsov.

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