Constantin Shramov
We prove that a nodal quartic threefold $X$ containing no planes is $Q$-factorial provided that it has not more than 12 singular points, with the exception of a quartic with exactly 12 singularities containing a quadric surface. We give some geometrical constructions related to the latter quartic.
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We obtain upper bounds on the number of singular points of factorial terminal Fano threefolds.
We define and calculate the weighted multiplicities of non Gorenstein terminal singularities on threefolds and some quotient singularities. We improve freeness conditions on threefolds.
Log Sarkisov Program for strongly $\mathbb{Q}$-factorial compact Kähler threefolds
该研究为强$\mathbb{Q}$-factorial紧Kähler三维簇建立对数Sarkisov纲领，解决了该类簇的双有理几何核心问题。
On maximally non-factorial nodal Fano threefolds
本文研究非阶乘节点 Fano 三维簇，完成了对恰含 1 个节点且类群秩为 2 情形的分类。
We prove the $\mathbb{Q}$-factoriality of a nodal hypersurface in $\mathbb{P}^{4}$ of degree $n$ with at most ${\frac{(n-1)^{2}}{4}}$ nodes and the $\mathbb{Q}$-factoriality of a double cover of $\mathbb{P}^{3}$ branched over a nodal surface of degree $2r$ with at most ${\frac{(2r-1)r}{3}}$ nodes.
The $\mathbb{Q}$-factoriality of a nodal quartic 3-fold implies its non-rationality. We prove that a nodal quartic 3-fold with at most 8 nodes is $\mathbb{Q}$-factorial, and we show that a nodal quartic 3-fold with 9 nodes is not $\mathbb{Q}$-factorial if and only if it contains a plane. However, there are non-rational non-$\mathbb{Q}$-factorial nodal quartic 3-folds in $\mathbb{P}^4$. In particular, we prove the non-rationality of a general non-$\mathbb{Q}$-factorial nodal quartic 3-fold that contains either a plane or a smooth del Pezzo surface of degree 4.
The paper explores the birational geometry of terminal quartic 3-folds. In doing this I develop a new approach to study maximal singularities with positive dimensional centers. This allows to determine the pliability of a Q-factorial quartic with ordinary double points, and it shows the importance of Q-factoriality in the context of birational geometry of uniruled 3-folds
For every smooth quartic threefold, we classify all pencils on it whose general element is an irreducible surface birational to a smooth surface of Kodaira dimension zero.
The affine cancellation problem, which asks whether complex affine varieties with isomorphic cylinders are themselves isomorphic, has a positive solution for two dimensional varieties whose coordinate rings are unique factorization domains, in particular for the affine plane, but counterexamples are found within normal surfaces Danielewski surfaces and factorial threefolds of logarithmic Kodaira dimension equal to 1. The latter are therefore remote from the affine three-space, the first unknown case where the base of one cylinder is an affine space. Locally trivial Ga-actions play a significan…
The quartic hypersurfaces in P^4 invariant under the standard representation of S_6 form a linear pencil. We prove that a general member of this pencil is not rational.
The automorphism groups of certain factorial complex affine threefolds admitting locally trivial actions of the additive group are determined. As a consequence new counterexamples to a generalized cancellation problem are obtained.
We study Nash valuations and essential valuations of terminal threefolds of type $cA/r$. If $r=1$ or the given threefold is $\mathbb Q$-factorial, then all the Nash valuations and essential valuations can be completely described. We construct non-Gorenstein or non-$\mathbb Q$-factorial counter examples for the Nash problem.
We give a bound on the minimal number of singularities of a nodal projective complete intersection threefold which contains a smooth complete intersection surface that is not a Cartier divisor.
We prove that the quartic threefolds defined by $$ \sum_{i=0}^{5}x_i=\sum_{i=0}^{5}x_i^4-t\left(\sum_{i=0}^{5}x_i^2\right)^2=0 $$ in $\mathbb{P}^5$ are rational for $t=\frac{1}{6}$ and $t=\frac{7}{10}$.
Let $X$ be a hypersurface in $\mathbb{P}^{4}$ of degree $d$ that has at most isolated ordinary double points. We prove that $X$ is factorial in the case when $X$ has at most $(d-1)^{2}-1$ singular points.
We study certain moduli spaces of stable vector bundles of rank two on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles.
We study threefolds fibred by mirror quartic K3 surfaces. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the moduli space of mirror quartic K3 surfaces. This is then used to give a complete explicit description of all Calabi-Yau threefolds fibred by mirror quartic K3 surfaces. We conclude by studying the properties of such Calabi-Yau threefolds, including their Hodge numbers and deformation theory.
A remark on rational quartic curves in prime Fano threefolds of degree $22$
利用度22的素Fano三维流形$V_{22}$与五次del Pezzo三维流形$V_5$之间的Sarkisov联系，证明$V_{22}$中有理四次曲线的Hilbert方案到射影空间$\mathbb{P}^4$存在一个一般2对1的有理映射。
Let $X\subset \mathbb{P}^4$ be a terminal factorial quartic $3$-fold. If $X$ is non-singular, $X$ is \emph{birationally rigid}, i.e. the classical MMP on any terminal $\mathbb{Q}$-factorial projective variety $Z$ birational to $X$ always terminates with $X$. This no longer holds when $X$ is singular, but very few examples of non-rigid factorial quartics are known. In this article, we first bound the local analytic type of singularities that may occur on a terminal factorial quartic hypersurface $X\subset \mathbb{P}^4$. A singular point on such a hypersurface is either of type $cA_n$ ($n\geq 1$…
We prove the factoriality of the following nodal threefolds: a complete intersection of hypersurfaces $F$ and $G\subset\mathbb{P}^{5}$ of degree $n$ and $k$ respectively, where $G$ is smooth, $|\mathrm{Sing}(F\cap G)|\leqslant(n+k-2)(n-1)/5$, $n\geqslant k$; a double cover of a smooth hypersurface $F\subset\mathbb{P}^{4}$ of degree $n$ branched over a surface that is cut out on $F$ by a hypersurface $G$ of degree $2r\geqslant n$, and $|\mathrm{Sing}(F\cap G)|\leqslant(2r+n-2)r/4$.
We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all $S_6$-invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman-Edge pencil. As an application, we check that $S_6$-invariant three-dimensional quartics are unirational, obtain new proofs of rationality of four special quartics among them and irrationality of the others, and describe their Weil divisor class grou…
This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not Q-factorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, Weil non-Cartier divisors are generated by "topological traces " of K-negative extremal contractions on X. One can show, as an application of these methods, that a number of families of non-factorial terminal Gorenstein Fano 3-folds are rational. In particular, I give some examples of rational quartic hypersurfaces wi…
Explicit birational geometry of determinantal quartic 3-folds
研究$\mathbb{P}^4$中一般线性行列式四次三维流形的双有理几何，发现其族包含有理$\mathbb{Q}$-factorial成员并可特化为有理非$\mathbb{Q}$-factorial双二次曲面，且几何由$\mathbb{P}^3$中10次11亏格曲线控制。
Quartic curves in the quintic del Pezzo threefold
本文证明五次 del Pezzo 三维簇 X5 上有理四次曲线的 Hilbert 概形 H4(X5) 同构于 X5 上直线的 Hilbert 概形上的 Grassmannian 丛，因此光滑且不可约。
In this paper we classify nodal rational non-$\mathbb{Q}$-factorial del Pezzo threefolds of degree 2 which can be $G$-birationally rigid for some subgroup $G\subset \operatorname{Aut}(X)$.
Degenerations of elliptic quartics and K-moduli of Fano threefolds
本文研究经两次爆破构造的法诺三维簇的K-模空间，将其同构于带CM线丛线性化的VGIT商空间，分类了该形变族的K-(半/多)稳定成员，相关证明用到希尔伯特概型几何等新要素。
It is well-known that a nonsingular minimal cubic surface is birationally rigid; the group of its birational selfmaps is generated by biregular selfmaps and birational involutions such that all relations between the latter are implied by standard relations between reflections on an elliptic curve. It is also known that a factorial nodal quartic threefold is birationally rigid and its group of birational selfmaps is generated by biregular ones and certain birational involutions. We prove that all relations between these involutions are implied by standard relations on elliptic curves, complete …
Moduli space of genus one curves on quartic and quintic del Pezzo threefolds
本文研究四次和五次 del Pezzo 三维簇上光滑亏格一曲线的空间，描述了 Kontsevich 模空间的不可约分支并分类了从一般椭圆曲线出发的态射空间，验证了复数域上所有四次和五次 del Pezzo 三维簇的几何 Manin 猜想。
We study local, global and local-to-global properties of threefolds with certain singularities. We prove criteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincaré duality. We relate the topological Euler characteristic of elliptic Calabi-Yau threefolds with $\mathbb Q$-factorial terminal singularities to dimensions of Lie algebras and certain representations, Milnor and Tyurina numbers and other birational invariants of an elliptic fibration. We give an interpretation in terms of complex deformations. We state a conjecture on the…
We study the mixed Hodge structure on the third homology group of a threefold which is the double cover of projective three-space ramified over a quartic surface with a double conic. We deal with the Torelli problem for such threefolds.
We prove that the degree of Fano threefolds with terminal Q-factorial singularities and Picard number one is at most 125/2 and the bound is sharp.
We classify the non-toric, $\mathbb Q$-factorial, Gorenstein, log terminal Fano threefolds of Picard number one that admit an effective action of a two-dimensional algebraic torus.
In this paper, we prove a Clifford type inequality for the curve $X_{2,2,2,4}$, which is the intersection of a quartic and three general quadratics in $\mathbb{P}^5$. We thus prove a stronger Bogomolov-Gieseker inequality for characters of stable vector bundles and stable objects on $X_{2,4}$. Applying the scheme proposed by Bayer, Bertram, Macrì, Stellari and Toda, we can construct an open subset of Bridgeland stability conditions on $X_{2,4}$.
In this paper we study ACM vector bundles $\E$ of rank $k \geq 3$ on hypersurfaces $X_r \subset\Pj^4$ of degree $r \geq 1$. We consider here mainly the case of degree $r = 4$, which is the first unknown case in literature. Under some natural conditions for the bundle $\E$ we derive a list of possible Chern classes $(c_1,c_2,c_3)$ which may arise in the cases of rank $k=3$ and $k=4$, when $r=4$. For some cases among these we give the corresponding examples, the existence of all the other cases remaining under question.
We prove birational rigidity and calculate the group of birational automorphisms of a nodal Q-factorial double cover $X$ of a smooth three-dimensional quadric branched over a quartic section. We also prove that $X$ is Q-factorial provided that it has at most 11 singularities; moreover, we give an example of a non-Q-factorial variety of this type with 12 simple double singularities.
A Finiteness Theorem for Quartic K3-Fibred Calabi--Yau Threefolds in Scrolls
本文针对堆卷中以四次K3曲面为纤维的Calabi-Yau三维簇，应用Reid-Shepherd-Barron-Tai准则分类了n=1,2,3的情况，共得到14个形变族，补充了相关显式分类数据。
We study two rational Fano threefolds with an action of the icosahedral group $\mathfrak{A}_5$. The first one is the famous Burkhardt quartic threefold, and the second one is the double cover of the projective space branched in the Barth sextic surface. We prove that both of them are $\mathfrak{A}_5$-Fano varieties that are $\mathfrak{A}_5$-birationally superrigid. This gives two new embeddings of the group $\mathfrak{A}_5$ into the space Cremona group.
One-nodal Fano threefolds with Picard number one
分类了Picard数为1的光滑Fano三维簇的所有1-节点退化，并揭示了其与更高Picard秩的光滑Fano三维簇及完全交簇投影之间的几何关系。
The quartic threefold is symplectically irrational
本文证明光滑四次三维流形是辛非有理的，即不能通过一系列辛爆破、爆缩和形变与射影空间相关联，从而恢复了Iskovskikh-Manin的经典代数非有理结果，证明中使用了Iritani的工作建立了沿辛爆破的量子上同调分解定理。
Derived categories of quartic double fivefolds
该研究通过扭曲的Calabi-Yau三重线构造了四次双五重线的导出范畴，并验证了Kuznetsov的有理性猜想和Reid的非交换幻想。
We show the validity of the relative dlt MMP over Q-factorial threefolds in all characteristics p>0. As a corollary, we generalise many recent results to low characteristics including: $W\mathcal{O}$-rationality of klt singularities, inversion of adjunction, and normality of divisorial centres up to a universal homeomorphism.
For a projective variety $Z$ and for any integer $p$, define the $p$-th Néron-Severi group $NS_p(Z)$ of $Z$ as the image of the cycle map $A_{p}(Z)\to H_{2p}(Z; \mathbb{C})$. Now let $X\subset \Ps^{2m+1}$ ($m\geq 1$) be a projective variety of dimension $2m-1$, with isolated singularities, complete intersection of a smooth hypersurface of degree $k$, with a hypersurface of degree $n>max\{k, 2m+1\}$, and let $F$ be a general hypersurface of degree $n$ containing $X$. We prove that the natural map $NS_m(X)\to NS_m(F)$ is surjective, and that if $dim NS_m(F)=1$ then $dim NS_m(X)=1$. In particular…
We study twists of the Burkhardt quartic threefold over non-algebraically closed base fields of characteristic different from 2,3,5. We show they all admit quartic models in projective four-space. We identify a Galois-cohomological obstruction that measures if a given twist is birational to a moduli space of abelian varieties. This obstruction has implications for the rational points on these varieties. As a result, we see that all possible 3-level structures can be realized by abelian surfaces, whereas Kummer 3-level structures that group-theoretically may be admissible, may not be realizable…
An effective upper bound for Fano indices of canonical Fano threefolds, I
本文证明了具有孤立典范奇点的$\mathbb Q$-因子化弱Fano三维流形的$\mathbb Q$-Fano指数上界为61。
Over a field $k$, we study rational UFDs of finite transcendence degree $n$ over $k$. We classify such UFDs $B$ when $n=2$, $k$ is algebraically closed, and $B$ admits a positive $\mathbb{Z}$-grading, showing in particular that $B$ is affine over $k$. We also consider the Russell cubic threefold over $\mathbb{C}$, and the Asanuma threefolds over a field of positive characterstic, showing that these threefolds admit no elliptic $\mathbb{G}_m$-action. Finally, we show that, if $X$ is an affine $k$-variety and $X\times\mathbb{A}^m_k\cong_k\mathbb{A}^{n+m}_k$, then $X\cong_k\mathbb{A}^n_k$ if and …
Gorenstein Fano threefolds of Picard number one with a $\mathbb{K}^*$-action and maximal orbit quotient $\mathbb{P}_2$
本文分类了一类具有 $\mathbb{K}^*$ 作用和最大轨道商 $\mathbb{P}_2$ 的非环、$\mathbb{Q}$-因子、Gorenstein、Picard 数为一的 Fano 三维簇。
K-stable Fano threefolds of rank 2 and degree 28
本文证明了除明确例外情况外，所有Picard秩2且次数为28的光滑Fano三维流形均为K-多稳定，并对光滑二次三维流形中有理正规四次曲线的法丛进行了分类。
On the MMP for rank one foliations on threefolds
本文证明 Q-因子射影 klt 三维簇上秩一对数典范叶状配对存在翻转，从而推广了三维秩一叶状结构极小模型的存在性结果。
Arithmetic aspects of the Burkhardt quartic threefold
本文研究 Burkhardt 四次三维簇的算术性质，证明其在特征非 3 域上有理、计算有限域 zeta 函数，并显式给出 genus 2 泛曲线模型及 j-平面与 3 阶子群、Hesse 铅笔的关系。
We prove that the general quartic double solid with $k\leq 7$ nodes does not admit a Chow theoretic decomposition of the diagonal, or equivalently has a nontrivial universal ${\rm CH}_0$ group. The same holds if we replace in this statement "Chow theoretic" by "cohomological". In particular, it is not stably rational. We also prove that the general quartic double solid with seven nodes does not admit a universal codimension 2 cycle parameterized by its intermediate Jacobian, and even does not admit a parametrization with rationally connected fibres of its Jacobian by a family of 1-cycles. This…
Desmic quartic surfaces in arbitrary characteristic
研究Desmic四次曲面作为椭圆曲线自积的Kummer曲面的双有理模型，在任意特征下推广其经典几何，并证明关联的三次线丛是有34个节点的有理Fano三维流形。
$\mathbb{Q}$-Fano threefolds of Fano index thirteen
本研究证明了 Picard 等级为 1 且 Fano 索引为 13 的 Fano 三重曲面为特定加权超曲面
We prove that up to birational equivalence, there exists only a finite number of families of Calabi-Yau threefolds (i.e. a threefold with trivial canonical class and factorial terminal singularities) which have an elliptic fibration to a rational surface. This strengthens a result of B. Hunt that there are only a finite number of possible Euler characteristics for such threefolds.
We consider the Noether-Lefschetz problem for surfaces in Q-factorial normal 3-folds with rational singularities. We show the existence of components of the Noether-Lefschetz locus of maximal codimension, and that there are indeed infinitely many of them. Moreover, we show that their union is dense in the natural topology.
Birational rigidity of quartic three-folds with a double point of rank 3
本文证明复射影空间中仅含秩3二重点的一般四次三重流形具有双有理刚性，其双有理自映射群为25个2阶循环群的自由积，并由此推出刚性阶乘四次流形补集的余维数至少为3。
Let $Z$ be the quotient of the Siegel modular threefold $\mathcal{A}^{\rm sa}(2,4,8)$ which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple $F_Z$ of theta constants which is in turn known to be a Klingen type Eisenstein series of weight 3 should be related to a holomorphic differential $(2,0)$-form on $Z$. The variety $Z$ is birationally equivalent to the tangent cone of Fermat quartic surface in the title. In this paper we first compute the L-function of two smooth resolutions of $Z$. One of these, denoted by $W$, is a kind of Igusa compactification such…
On K-moduli of quartic threefolds
本文研究四次三维簇 K-模紧化，证明加权完全交 $X_{2,2,4}$ 的 K-多重稳定性及其在模空间闭包中的边界现象。
Rational quartic curves in the Mukai-Umemura variety
本文研究了Mukai-Umemura簇中的有理四次曲线，证明了其Hilbert概形的光滑性，并利用Białynicki-Birula定理计算了该概形的Poincaré多项式。
We prove that a very general double cover of the projective four-space, ramified in a quartic threefold, is not stably rational.
We prove the boundedness theorem for Fano threefolds with log-terminal singularities of any fixed index. This is an improvement of our earlier result, where we required additionally that the variety is Q-factorial, with Picard number 1. The new ideas of the paper include the following. 1. Using Alexeev Minimal Model program with suitable boundary to find horizontal extremal contractions. 2. Using Kollár's effective Base Point Freeness theorem. 3. Using Kawamata's result on the length of extremal curves with suitable boundary to avoid gluing curves in some cases.
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