Fano Hypersurfaces in Positive Characteristic

2-Cycles on Higher Fano Hypersurfaces

Let F(X_d) be a smooth Fano variety of lines of a hypersurface X_d of degree d. In this paper, we prove the Griffiths group Griff_1(F(X_d)) is trivial if the hypersurface X_d is of 2-Fano type. As a result, we give a positive answer to a question of Professor Voisin about the first Griffiths groups of Fano varieties in some cases. Base on this result, we prove that CH_2(X_d)=\mathbb{Z} for a complex smooth $3$-Fano hypersurface X_d whose Fano variety of lines is smooth.

指数1的Fano三维超曲面的K-稳定性

K-stablity of Fano threefold hypersurfaces of index 1

通过给出delta不变量的下界，解决了Fano指数为1的拟光滑Fano三维超曲面的K-稳定性问题，使用了Abban和Zhuang引入的方法计算Fano三维超曲面旗上的delta不变量下界。

Stable rationality of orbifold Fano threefold hypersurfaces

We determine the rationality of very general quasismooth Fano 3-fold weighted hypersurfaces completely and determine the stable rationality of them except for cubic 3-folds. More precisely we prove that (i) very general Fano 3-fold weighted hypersurfaces of index 1 or 2 are not stably rational except possibly for the cubic threefolds, (ii) among the 27 families of Fano 3-fold weighted hypersurfaces of index greater than 2, very general members of specific 7 families are not stably rational and the remaining 20 families consists of rational varieties.

Rational surfaces in index-one Fano hypersurfaces

We give the first evidence for a conjecture that a general, index-one, Fano hypersurface is not unirational: (i) a general point of the hypersurface is contained in no rational surface ruled, roughly, by low-degree rational curves, and (ii) a general point is contained in no image of a Del Pezzo surface.

Elliptic structures on weighted three-dimensional Fano hypersurfaces

We classify birational maps into elliptic fibrations of a general quasismooth hypersurface in $\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree $\sum_{i=1}^{4}a_{i}$ that has terminal singularities.

Birational rigidity of singular Fano hypersurfaces

We establish birational superrigidity for a large class of singular projective Fano hypersurfaces of index one. In the special case of isolated singularities, our result applies for instance to: (1) hypersurfaces with semi-homogeneous singularities of multiplicity asymptotically bounded by twice the square root of the dimension of the hypersurface, (2) hypersurfaces with isolated singularities whose Tyurina numbers satisfy a similar bound, and (3) hypersurfaces with isolated singularities whose dual variety is a hypersurface of degree sufficiently close to the expected degree.

On the degree of Fano schemes of linear subspaces on hypersurfaces

In this paper we propose and prove an explicit formula for computing the degree of Fano schemes of linear subspaces on general hypersurfaces. The method used here is based on the localization theorem and Bott's residue formula in equivariant intersection theory.

Halphen Pencils on Weighted Fano Threefold Hypersurfaces

We classify all pencils on a general weighted hypersurface in $\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree $\sum_{i=1}^{4}a_{i}$ whose general members are surfaces of Kodaira dimension zero.

Birationally rigid Fano hypersurfaces with isolated singularities

It is proved that a general Fano hypersurface of index 1 (in the projective space) with isolated singularities of general position is birationally rigid. Therefore it cannot be fibered into uniruled varieties of a smaller dimension by a rational map. The group of birational self-maps is either trivial or cyclic of order two.

Birationally rigid Fano hypersurfaces

We prove that a smooth Fano hypersurface $V=V_M\subset{\Bbb P}^M$, $M\geq 6$, is birationally superrigid. In particular, it cannot be fibered into uniruled varieties by a non-trivial rational map and each birational map onto a minimal Fano variety of the same dimension is a biregular isomorphism. The proof is based on the method of maximal singularities combined with the connectedness principle of Shokurov and Koll\' ar.

Fano hypersurfaces in weighted projective 4-spaces

We determine the complete list of anticanonically embedded quasi smooth log Fano 3-folds in weighted projective 4-spaces. This implies that the Reid-Fletcher list of 95 types of anticanonically embedded quasi smooth terminal Fano threefolds in weighted projective 4-spaces is complete. We also prove that there are only finitely many families of quasi smooth Calabi-Yau hypersurfaces in weighted projective spaces of any given dimension.

Rational endomorphisms of Fano hypersurfaces

We show that the degrees of rational endomorphisms of very general complex Fano and Calabi-Yau hypersurfaces satisfy certain congruence conditions by specializing to characteristic p. As a corollary we show that very general n-dimensional hypersurfaces of degree $d\ge 5\lceil (n+3)/6\rceil$ are not birational to elliptic fibrations. A key part of the argument is to resolve singularities of general p-cyclic covers in mixed characteristic p.

双有理超刚性Fano三维加权超曲面的K-稳定性

K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces

该研究证明指数为1的拟光滑Fano三维加权超曲面的α不变量≥1/2，结合已有双有理超刚性与K-稳定性的关系结论，推导出这类双有理超刚性曲面的K-稳定性。

Fano varieties and linear sections of hypersurfaces

Under a hypothesis on $k$, $d$ and $n$ that is almost the best possible, we prove that for every smooth degree $d$ hypersurface in $P^n$, the $k$-plane sections dominate the moduli space of degree $d$ hypersurface in $P^k$. Using this we prove rational simple connectedness of every smooth degree $d$ hypersurface in $P^n$, under a suitable hypothesis on $d$ and $n$ (previous results were only for general hypersurfaces).

The degree of irrationality of hypersurfaces in various Fano varieties

The purpose of this paper is to compute the degree of irrationality of hypersurfaces of sufficiently high degree in various Fano varieties: quadrics, Grassmannians, products of projective space, cubic threefolds, cubic fourfolds, and complete intersection threefolds of type (2,2). This extends the techniques of Bastianelli, De Poi, Ein, Lazarsfeld, and the second author who computed the degree of irrationality of hypersurfaces of sufficiently high degree in projective space. A theme in the paper is that the fibers of low degree rational maps from the hypersurfaces to projective space tend to l…

Troisième groupe de cohomologie non ramifiée des hypersurfaces de Fano

We establish the vanishing of the third unramified cohomology group for many types of Fano hypersurfaces in projective space over an algebraically closed field of arbitrary characteristic, and over a finite field. For cubic hypersurfaces over a finite field, the case of fourfolds remains open. --- Sur un corps algébriquement clos et sur un corps fini, on établit de nouveaux résultats d'annulation pour la cohomologie non ramifiée de degré 3 pour de nombreux types d'hypersurfaces de Fano. Le cas des hypersurfaces cubiques de dimension 4 sur un corps fini reste ouvert.

The Fano variety of lines and rationality problem for a cubic hypersurface

We find a relation between a cubic hypersurface $Y$ and its Fano variety of lines $F(Y)$ in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational to a Hilbert scheme of two points on a K3 surface; in particular, general cubic fourfold is irrational.

High-pliability Fano hypersurfaces

We show that five of Reid's Fano 3-fold hyperurfaces containing at least one compound Du Val singularity of type $cA_n$ have pliability at least two. The two elements of the pliability set are the singular hypersurface itself, and another non-isomorphic Fano hypersurface of the same degree, embedded in the same weighted projective space, but with different compound Du Val singularities. The birational map between them is the composition of two birational links initiated by blowing up two Type I centres on a codimension 4 Fano 3-fold of $\mathbb{P}^2 \times \mathbb{P}^2$-type having Picard rank…

Fano hypersurfaces with arbitrarily large degrees of irrationality

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index e, then the degree of irrationality of a very general complex Fano hypersurface of index e and dimension n is bounded from below by a constant times $\sqrt{n}$. To our knowledge this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic p argument which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way we show that in a family of varieties, …

通过plt标志和凸几何研究Fano加权超曲面的K稳定性

K-stability of Fano weighted hypersurfaces via plt flags and convex geometry

本文通过结合birational和凸几何技术，证明了特定条件下的加权Fano超曲面的K稳定性，并构造了K不稳定的例子。

Kontsevich spaces of rational curves on Fano hypersurfaces

We investigate the spaces of rational curves on a general hypersurface. In particular, we show that for a general degree $d$ hypersurface in $\mathbb{P}^n$ with $n \geq d+2$, the space $\overline{\mathcal{M}_{0,0}}(X,e)$ of degree $e$ Kontsevich stable maps from a rational curve to $X$ is an irreducible local complete intersection stack of dimension $e(n-d+1)+n-4$.

Stable rationality of index one Fano hypersurfaces containing a linear space

We prove that a very general complex hypersurface of degree $n+1$ in $\mathbb{P}^{n+1}$ containing an $r$-plane with multiplicity $m$ is not stably rational for $n \ge 3$, $m, r > 0$ and $n \ge m+r$. We also investigate failure of stable rationality of a very general hypersurface of degree $n+1$ in $\mathbb{P}^{n+1}$ admitting several isolated ordinary double points.

双有理刚性的Fano 3-重超曲面

Birationally solid Fano 3-fold hypersurfaces

本文基于Picard数为1的Fano簇的双有理刚性定义，完成了拟光滑双有理刚性Fano 3-重加权超曲面的分类。

Effective birational rigidity of Fano double hypersurfaces

We prove birational superrigidity of Fano double hypersurfaces of index one with quadratic and multi-quadratic singularities, satisfying certain regularity conditions, and give an effective explicit lower bound for the codimension of the set of non-rigid varieties in the natural parameter space of the family. The lower bound is quadratic in the dimension of the variety. The proof is based on the techniques of hypertangent divisors combined with the recently discovered $4n^2$-inequality for complete intersection singularities.

Fano超曲面中的自由曲线必须具有高次数

Free curves in Fano hypersurfaces must have high degree

本文证明在正特征域中，Fano超曲面中自由有理曲线的最小次数无法由n的线性函数界定，并给出了超线性的界。

四维二次超曲面Q^4上的秩二弱法诺丛

Rank-two weak Fano bundles on a four-dimensional quadric hypersurface $Q^4$

本文对四维二次超曲面Q^4上的秩2弱法诺丛分类，经线丛扭转后这类丛分为三类，含文献[APW94]构造的特定稳定丛。

K3 型加权四维 Fano 超曲面

Weighted four-dimensional Fano hypersurfaces of K3 type

本文证明在奇点轨迹维数至多为一的假设下，K3 型加权 Fano 四维超曲面族仅有有限多个，并给出完整列表及有理性和奇点分析。

Birational rigidity and K-stability of Fano hypersurfaces with ordinary double points

Extending previous results, we prove that for $n \ge 5$ all hypersurfaces of degree $n+1$ in ${\mathbb P}^{n+1}$ with isolated ordinary double points are birational superrigid and K-stable, hence admit a weak Kähler--Einstein metric.

The Curve Exclusion Theorem for elliptic and K3 fibrations birational to Fano 3-fold hypersurfaces

The theorem referred to in the title is a technical result that is needed for the classification of elliptic and K3 fibrations birational to Fano 3-fold hypersurfaces in weighted projective space. We present a complete proof of the Curve Exclusion Theorem, which first appeared in the author's unpublished PhD thesis [Ry02] and has since been relied upon in solutions to many cases of the fibration classification problem. We give examples of these solutions and discuss them briefly.

Birational geometry of singular Fano hypersurfaces of index two

For a Zariski general (regular) hypersurface $V$ of degree $M$ in the $(M+1)$-dimensional projective space, where $M$ is at least 16, with at most quadratic singularities of rank at least 13, we give a complete description of the structures of rationally connected (or Fano-Mori) fibre space: every such structure over a positive-dimensional base is a pencil of hyperplane sections. This implies, in particular, that $V$ is non-rational and its groups of birational and biregular automorphisms coincide. The set of non-regular hypersurfaces has codimension at least $\frac12(M-11)(M-10)-10$ in the na…

Families of Calabi-Yau hypersurfaces in $\mathbb Q$-Fano toric varieties

We provide a sufficient condition for a general hypersurface in a $\mathbb Q$-Fano toric variety to be a Calabi-Yau variety in terms of its Newton polytope. Moreover, we define a generalization of the Berglund-Hübsch-Krawitz construction in case the ambient is a $\mathbb Q$-Fano toric variety with torsion free class group and the defining polynomial is not necessarily of Delsarte type. Finally, we introduce a duality between families of Calabi-Yau hypersurfaces which includes both Batyrev and Berglund-Hübsch-Krawitz mirror constructions. This is given in terms of a polar duality between pairs …

Fano hypersurfaces with no finite order birational automorphisms

We use the specialization homomorphism for the birational automorphism group to study finite order birational automorphisms. For a family of varieties over a DVR, we prove that a birational automorphism of order coprime to the residue characteristic cannot specialize to the identity. As an application, we show that very general $n$-dimensional hypersurfaces of degree $d \geq 5 \lceil (n+3)/6 \rceil$ have no finite order birational automorphisms.

Balanced rational curves and minimal rational connectedness of Fano hypersurfaces

On a general Fano hypersurface in projective space, we determine for infinitely many $k$ the minimal degree $e$ of a rational curve through a general collection of $k$ points. In the case of a hypersurface of index 1, our results hold for all $k\geq 1$. In an appendix, M.C. Chang proves an arithmetical result which implies that in the case of index $>1$, the density of the set of curve degrees $e$ covered by our method is approximately $\frac{(n-d)(d-\frac{5}{2})}{(n-2)d}$.

On birational superrigidity and conditional birational superrigidity of certain Fano hypersurfaces

We prove birational superrigidity of every hypersurface of degree N in P^N with singular locus of dimension s, under the assumption that N is at least 2s+8 and it has only quadratic singularities of rank at least N-s. Combined with the results of I. A. Chel'tsov and T. de Fernex, this completes the list of birationally superrigid singular hypersurfaces with only ordinary double points except in dimension 4 and 6. Further we impose an additional condition on the base locus of a birational map to a Mori fiber space. Then we prove conditional birational superrigidity of certain smooth Fano hypers…

加权超曲面Fano三维簇的自同构群与柱性

Automorphism groups and cylindricity of weighted hypersurface Fano threefolds

本文针对拟光滑终端加权超曲面Fano三维簇剩余的20个形变族，研究其每个成员的柱性与完全自同构群，并探讨这些族的形式的柱性，补充了相关分类研究的空白。

Dimension of the space of conics on Fano hypersurfaces

R. Beheshti showed that, for a smooth Fano hypersurface $X$ of degree $\leq 8$ over the complex number field $\mathbb{C}$, the dimension of the space of lines lying in $X$ is equal to the expected dimension. We study the space of conics on $X$. In this case, if $X$ contains some linear subvariety, then the dimension of the space can be larger than the expected dimension. In this paper, we show that, for a smooth Fano hypersurface $X$ of degree $\leq 6$ over $\mathbb{C}$, and for an irreducible component $R$ of the space of conics lying in $X$, if the $2$-plane spanned by a general conic of $R$…

Birational geometry of Fano hypersurfaces of index two

We prove that every non-trivial structure of a rationally connected fibre space (and so every structure of a Mori-Fano fibre space) on a general (in the sense of Zariski topology) hypersurface of degree $M$ in the $(M+1)$-dimensional projective space for $M\geq 14$ is given by a pencil of hyperplane sections. In particular, the variety $V$ is non-rational and its group of birational self-maps coincide with the group of biregular automorphisms and is therefore trivial. The proof is based on the techniques of the method of maximal singularities and the inversion of adjunction.

Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces

Let $X\subset ¶^n$ be a possibly singular hypersurface of degree $d\le n$, defined over a finite field $k$. We show that the diagonal, suitably interpreted, is decomposable. This gives a proof that the eigenvalues of the Frobenius action on its $\ell$-adic cohomology $H^i(\bar{X}, \Q_\ell)$, for $\ell \neq {\rm char}(k)$, are divisible by $q$, without using the result on the existence of rational points by Ax and Katz.

Derived categories of flips and cubic hypersurfaces

A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by describing the complement. As an application, we can lift the "quadratic Fano correspondence" (due to Galkin-Shinder) in the Grothendieck ring of varieties between a smooth cubic hypersurface, its Fano variety of lines, and its Hilbert square, to a semiorthogonal decomposition. We also show that the Hilbert square of a cubic hypersurface of dimension at leas…

A note on rational curves on general Fano hypersurfaces

We show the Kontsevich space of rational curves of degree at most roughly $\frac{2-\sqrt{2}}{2}n$ on a general hypersurface $X\subset \mathbb{P}^n$ of degree $n-1$ is equidimensional of expected dimension and has two components: one consisting generically of smooth, embedded rational curves and the other consisting of multiple covers of a line. This proves more cases of a conjecture of Coskun, Harris, and Starr and shows the Gromov-Witten invariants in these cases are enumerative.

Fano超曲面上曲线的极化插值与正规假设

Polarized interpolation and normal postulation for curves on Fano hypersurfaces

本文证明Fano超曲面上充分大次数曲线的法丛与余法丛具有良好假设或自然上同调，从而建立极化插值性质。

法诺超曲面的纤维化亏格

The fibering genus of Fano hypersurfaces

本文拓展Kollár的思路，通过退化到正特征情形并利用Tate亏格变化公式，证明存在纤维化亏格可任意大的Fano超曲面，首次在Fano范围内研究簇的纤维化亏格问题。

由Fano多面体导出的环面$K3$超曲面的椭圆纤维化与镜像对称

Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes

本文确定了由Fano多面体导出的环面$K3$超曲面的Néron-Severi格，通过引入特定椭圆纤维化并证明其生成元，从而证明了Dolgachev猜想。

On a Conjecture on the Variety of Lines on a Fano Complete Intersection

The Debarre-de Jong conjecture predicts that the Fano variety of lines on a smooth Fano hypersurface in $\mathbb{P}^n$ is always of the expected dimension. We generalize this conjecture to the case of Fano complete intersections and prove that for a Fano complete intersection $X\subset \mathbb{P}^n$ of hypersurfaces whose degrees sum to at most 7, the Fano variety of lines on $X$ has the expected dimension.

Hilbert schemes of rational curves on Fano hypersurfaces

In this paper we try to further explore the linear model of the moduli of rational maps. Our attempt yields following results. Let $X\subset \mathbf P^n$ be a generic hypersurface of degree $h$. Let $R_d(X, h)$ denote the open set of the Hilbert scheme parameterizing irreducible rational curves of degree $d$ on $X$. We obtain that (1) If $4\leq h\leq n-1$, $R_d(X, h)$ is an integral, local complete intersection of dimension \begin{equation} (n+1-h)d+n-4. \end{equation} (2) If furthermore $(h^2-n)d+h\leq 0$ and $h\geq 4$, in addition to part (1), $R_d(X, h)$ is also rationally connected.

Interpolation of curves on Fano hypersurfaces

On a general hypersurface of degree $d\leq n$ in $\mathbb P^n$ or $\mathbb P^n$ itself, we prove the existence of curves of any genus and high enough degree depending on the genus passing through the expected number $t$ of general points or incident to a general collection of subvarieties of suitable codimensions. In some cases we also show that the family of curves through $t$ fixed points has general moduli as family of $t$-pointed curves. These results imply positivity of certain intersection numbers on Kontsevich spaces of stable maps. An arithmetical appendix by M. C. Chang descibes the s…

光滑法诺超曲面的爆破、其双有理几何与除子稳定性

Blowups of smooth Fano hypersurfaces, their birational geometry and divisorial stability

针对光滑法诺超曲面$X$沿子超曲面$Γ$的爆破$Y$，构造性证明$Y$是Mori梦想空间，描述其Mori chamber分解与双有理模型，分类使$Y$为法诺流形的$X$和$Γ$并启动K稳定性研究。

Torsion orders of Fano hypersurfaces

We find new lower bounds on the torsion orders of very general Fano hypersurfaces over (uncountable) fields of arbitrary characteristic. Our results imply that unirational parametrizations of most Fano hypersurfaces need to have enormously large degree. Our results also hold in characteristic two, where they solve the rationality problem for hypersurfaces under a logarithmic degree bound, thereby extending a previous result of the author from characteristic different from two to arbitrary characteristic.

Birational geometry of Fano double covers

We prove divisorial canonicity of Fano double hypersurfaces of general position.

Four-dimensional projective orbifold hypersurfaces

We classify four-dimensional quasismooth weighted hypersurfaces with small canonical class, and verify a conjecture of Johnson and Kollar on infinite series of quasismooth hypersurfaces with anticanonical hyperplane section in the case of fourfolds. By considering the quotient singularities that arise, we classify those weighted hypersurfaces that are canonical, Calabi-Yau, and Fano fourfolds. We also consider other classes of hypersurfaces, including Fano hypersurfaces of index greater than 1 in dimensions 3 and 4.

随机 Fano 超曲面上有理点的分布

Distribution of rational points on random Fano hypersurfaces

本文证明了随机 Fano 超曲面上有理点平均数量的渐近公式，以及容许 adelic 点的超曲面数量的渐近公式。

Weighted Fano threefold hypersurfaces

We study birational transformations into elliptic fibrations and birational automorphisms of quasismooth anticanonically embedded weighted Fano 3-fold hypersurfaces having terminal singularities classified by A.R. Iano-Fletcher, J. Johnson, J. Kollar, and M. Reid.

非--Fano环面簇中Calabi-Yau超曲面的热带周期

Tropical Periods for Calabi-Yau Hypersurfaces in non--Fano Toric Varieties

本文利用多面体描述非-Fano环面簇及其反典范Calabi-Yau超曲面，通过广义Duistermaat-Heckman测度从镜像流形周期证明$\widehatΓ$-猜想适用于非-Fano环境空间，扩展了早期工作。

Birational geometry of singular Fano varieties

We prove divisorial canonicity of Fano hypersurfaces and double spaces of general position with elementary singularities.