K-stable Fano 3-folds in the families 2.18 and 3.4
We prove that smooth Fano 3-folds in the families 2.18 and 3.4 are K-stable.
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arXiv subjects
Takashi Kishimoto
Publications and source records attributed to Takashi Kishimoto.
Cylinders in weighted Fano varieties
Cylinders in Fano varieties receives a lot of attentions recently from the viewpoints of birational geometry and unipotent geometry. In this article, we provide a survey of several known et new results concerning the anti-canonically polar cylindricity of quasi-smooth, well-formed weighted Fano complete intersections in weighted projective spaces.
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On K-stability of singular hyperelliptic Fano 3-folds
We study the K-stability of singular Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system is base-point-free but not very ample.
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Another view on smooth prime Fano threefolds of degree 22 with infinite automorphism groups
We give a self-contained alternative proof of the classification of smooth prime Fano threefolds of degree 22 with infinite automorphism groups established by Kuznetsov, Prokhorov and Shramov.
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Automorphism groups and cylindricity of weighted hypersurface Fano threefolds
It is well known that there are totally 130 deformation families of quasi-smooth terminal weighted hypersurface Fano threefolds and all members belonging to 95 families of Fano indices one are birationally rigid. Among remaining $35$ families, $15$ families have the property that general members are irrational, in particular, any of them has a finite automorphism group and is not cylindrical. In the present paper, we will observe the cylindricity and the full automorphism groups of every member in the remaining $20$ families. Moreover, we deal with the cylindricity of their forms.
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Smooth Fano 3-folds satisfying Condition (A)
A smooth variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We classify smooth Fano 3-folds that satisfy Condition (A).
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K-stability of Fano 3-folds in the World of Null-A
A variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We show that a smooth Fano 3-fold not satisfying Condition (A) is K-polystable unless it is contained in eight exceptional deformation families (seven of them consists of one smooth member, and one of them has one-parameter moduli).
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K-stability of pointless del Pezzo surfaces and Fano 3-folds
We explore connections between existence of $\Bbbk$-rational points for Fano varieties defined over $\Bbbk$, a subfield of $\mathbb{C}$, and existence of Kähler-Einstein metrics on their geometric models. First, we show that geometric models of del Pezzo surfaces with at worst quotient singularities defined over $\Bbbk\subset\mathbb{C}$ admit (orbifold) Kähler--Einstein metrics if they do not have $\Bbbk$-rational points. Then we prove the same result for smooth Fano 3-folds with 8 exceptions. Consequently, we explicitly describe several families of pointless Fano 3-folds whose geometric models admit Kähler-Einstein metrics. In particular, we obtain new examples of prime Fano 3-folds of genus $12$ that admit Kähler--Einstein metrics. Our result can also be used to prove existence of rational points for certain Fano varieties, for example for any smooth Fano 3-fold over $\Bbbk\subset\mathbb{C}$ whose geometric model is strictly K-semistable.
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Completions of the affine $3$-space into del Pezzo fibrations
We give constructions of completions of the affine $3$-space into total spaces of del Pezzo fibrations of every degree other than $7$ over the projective line. We show in particular that every del Pezzo surface other than $\mathbb{P}^{2}$ blown-up in one or two points can appear as a closed fiber of a del Pezzo fibration $π:X\to\mathbb{P}^{1}$ whose total space $X$ is a $\mathbb{Q}$-factorial threefold with terminal singularities which contains $\mathbb{A}^{3}$ as the complement of the union of a closed fiber of $π$ and a prime divisor $B_{h}$ horizontal for $π$. For such completions, we also give a complete description of integral curves that can appear as general fibers of the induced morphism $\barπ:B_{h}\to\mathbb{P}^{1}$.
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Toric G-solid Fano threefolds
We study toric G-solid Fano threefolds that have at most terminal singularities, where G is an algebraic subgroup of the normalizer of a maximal torus in their automorphism groups.
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K-stable divisors in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,2)$
We prove that every smooth divisor in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,2)$ is K-stable.
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Del Pezzo quintics as equivariant compactifications of vector groups
We study faithful actions with a dense orbit of abelian unipotent groups on quintic del Pezzo varieties over a field of characteristic zero. Such varieties are forms of linear sections of the Grassmannian of planes in a 5-dimensional vector space. We characterize which smooth forms admit these types of actions and show that in case of existence, the action is unique up to equivalence by automorphisms. We also give a similar classification for mildly singular quintic del Pezzo threefolds and surfaces.
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Completions of affine spaces into Mori fiber spaces with non-rational fibers
We describe a method to construct completions of affine spaces into total spaces of $\mathbb{Q}$-factorial terminal Mori fiber spaces over the projective line. As an application we provide families of examples with non-rational, birationally rigid and non-stably rational general fibers.
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Rees algebras of additive group actions
We establish basic properties of a sheaf of graded algebras canonically associated to every relative affine scheme $f : X \rightarrow S$ endowed with an action of the additive group scheme $\mathbb{G}_{ a,S}$ over a base scheme or algebraic space $S$, which we call the (relative) Rees algebra of the $\mathbb{G}_{ a,S}$-action. We illustrate these properties on several examples which played important roles in the development of the algebraic theory of locally nilpotent derivations and give some applications to the construction of families of affine threefolds with Ga-actions.
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Cylinders in Mori Fiber Spaces: forms of the quintic del Pezzo threefold
Motivated by the general question of existence of open A1-cylinders in higher dimensional pro-jective varieties, we consider the case of Mori Fiber Spaces of relative dimension three, whose general closed fibers are isomorphic to the quintic del Pezzo threefold V5 , the smooth Fano threefold of index two and degree five. We show that the total spaces of these Mori Fiber Spaces always contain relative A2-cylinders, and we characterize those admitting relative A3-cylinders in terms of the existence of certain special lines in their generic fibers.
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Deformations of $\mathbb{A}^1$-cylindrical varieties
An algebraic variety is called $\mathbb{A}^{1}$-cylindrical if it contains an $\mathbb{A}^{1}$-cylinder, i.e. a Zariski open subset of the form $Z\times\mathbb{A}^{1}$ for some algebraic variety Z. We show that the generic fiber of a family $f:X\rightarrow S$ of normal $\mathbb{A}^{1}$-cylindrical varieties becomes $\mathbb{A}^{1}$-cylindrical after a finite extension of the base. Our second result is a criterion for existence of an $\mathbb{A}^{1}$-cylinder in X which we derive from a careful inspection of a relative Minimal Model Program ran from a suitable smooth relative projective model of X over S.
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Equivariant extensions of Ga-torsors over punctured surfaces
Motivated by the study of the structure of algebraic actions the additive group on affine threefolds X, we consider a special class of such varieties whose algebraic quotient morphisms X $\rightarrow$ X//Ga restrict to principal homogeneous bundles over the complement of a smooth point of the quotient. We establish basic general properties of these varieties and construct families of examples illustrating their rich geometry. In particular, we give a complete classification of a natural subclass consisting of threefolds X endowed with proper Ga-actions, whose algebraic quotient morphisms $π$ : X $\rightarrow$ X//Ga are surjective with only isolated degenerate fibers, all isomorphic to the affine plane A 2 when equipped with their reduced structures.
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Cylinders in del Pezzo fibrations
We show that a del Pezzo fibration $π$ : V $\rightarrow$ W of degre d contains a vertical open cylinder, that is, an open subset whose intersection with the generic fiber of $π$ is isomorphic to $Z\times\mathbb{A}_{K}^{1}$ for some quasi-projective variety Z defined over the function field K of W , if and only if d $\ge$ 5 and $π$ : V $\rightarrow$ W admits a rational section. We also construct twisted cylinders in total spaces of threefold del Pezzo fibrations $π$ : V $\rightarrow$ P 1 of degree d $\le$ 4.
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