Ivan Cheltsov
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.
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We investigate the existence of complete intersection threefolds $X \subset \mathbb{P}^n$ with only isolated, ordinary multiple points and we provide some sufficient conditions for their factoriality.
Let X be a complete intersection of two hypersurfaces F_n and F_k in the projective space P^5 of degree n and k respectively with n >= k, such that the singularities of X are nodal and F_k is smooth. We prove that if the threefold X has at most (n+k-2)(n-1)-1 singular points, then it is factorial.
Q-factoriality and Hodge-Du Bois theory
该研究证明了正规射影簇及正规簇解析芽的$\text{Q}$-因子性缺陷的霍奇理论公式，其结果涵盖Samuel猜想局部解析版本、有理同调三维簇刻画及霍奇-杜布瓦数在翻转下的不变性等。
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.
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 prove the factoriality of a nodal hypersurface in $\mathbb{P}^{4}$ of degree $d$ that has at most $2(d-1)^{2}/3$ singular points, and factoriality of a double cover of $\mathbb{P}^{3}$ branched over a nodal surface of degree $2r$ having less than $(2r-1)r$ singular points.
We generalize Werner's defect formula for nodal hypersurfaces in $\mathbb P^{4}$ to the case of a nodal complete intersection threefold.
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.
We discuss the work of Corrado Segre on nodal cubic hypersurfaces with emphasis on the cases of 6-nodal and 10-nodal cubics. In particular, we discuss the Fano surface of lines and conic bundle structures on such threefolds. We review some of the current research in algebraic geometry related to Segre's work.
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。
On K-stability of one-nodal prime Fano threefolds of genus $12$
本文证明了亏格 $12$ 的一般一节点素 Fano 三维fold 是 K-多稳定的。
The Hasse principle for 9-nodal cubic threefolds
本文证明了在数域上具有9个奇点的三次三流形满足哈塞原理，解决了该类代数簇的局部-整体原则问题。
We study the Picard variety of the Fano surface of nodal and mildly cuspidal cubic threefolds in arbitrary characteristic by relating divisors on the Fano surface to divisors on the symmetric product of a curve of genus 4.
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。
We study the geometry and the period map of nodal complex prime Fano threefolds with index 1 and degree 10. We show that these threefolds are birationally isomorphic to Verra solids (hypersurfaces of bidegree $(2,2)$ in $ ¶^2\times ¶^2$). Using Verra's results on the period map for these solids and on the Prym map for double étale covers of plane sextic curves, we prove that the fiber of the period map for our nodal threefolds is birationally the union of two surfaces, for which we give several descriptions. This result is the analog in the nodal case of a result obtained in arXiv:0812.3670 fo…
Derived categories of nodal del Pezzo threefolds
本文完整回答了结点del Pezzo三维簇导出范畴的Kawamata型半正交分解存在性问题，证明了度数1至4无此分解而度数5存在，并利用从直线的投影构造了度数5的分解。
Nodal quintic del Pezzo threefolds and their derived categories
本文为节点五次 del Pezzo 三维簇构造川又型半正交分解，将其导出范畴分解为有限维代数的导出范畴，通过双有理映射到二次曲面纤维化实现。
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 $t=\frac{7}{10}$.
In this paper we show that a nodal complete intersection threefold $X$ in $\mathbb{P}^{3+c}$ with defect, but without induced defect, has at least $\sum_{i\leq j} (d_i-1)(d_j-1)$ nodes, provided either $c=2$ or $d_c>\sum_{i=1}^{c-1} d_i$ holds.
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 …
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 $|\mathrm{Sing}(F\cap G)|\leqslant(2r+n-2)r/4$.
Spinor modifications of conic bundles and derived categories of 1-nodal Fano threefolds
本文针对平坦二次曲线丛定义旋量修改构造新二次曲线丛，证明其偶克利福德代数Morita等价且导出范畴中基底导出范畴的正交补等价，并以非因子1-结点素法诺三维簇为例展示该技术，实现奇点范畴吸收。
The defect of a cubic threefold and applications to intermediate Jacobian fibrations
本文研究具有孤立奇点的三次三流形的缺陷不变量，利用过奇点的直线曲线拓扑数据计算缺陷，刻画混合霍奇结构，并证明非Q-阶乘性等价于包含平面或三次直纹面，进而联系到三次四流形的中间雅可比纤维化。
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.
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 $G\subset \operatorname{Aut}(X)$.
The main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds. Precisely, given $X$ a smooth projective threefold, $\E$ a rank-two vector bundle on $X$, $L$ a very ample line bundle on $X$ and $k \geq 0$, $δ>0 $ integers and denoted by $V= {\V}_δ ({\E} \otimes L^{\otimes k})$ the subscheme of ${\Pp}(H^0({\E} \otimes L^{\otimes k}))$ parametrizing global sections of ${\E} \otimes L^{\otimes k}$ whose zero-loci are irreducible and $δ$-nodal curves on $X$, we present a new cohomological description of the tangent space $T_{[s]}({\V}_δ ({\…
Double covers of smooth quadric threefolds with Artin-Mumford obstructions to rationality
本文分类了作为光滑二次三重流形双重覆盖且具有 Artin-Mumford 有理性障碍的节点 Fano 三重流形，证明其属于三个明确描述的族，其中两个族为首次发现。
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.
Let X be a (possibly nodal) K-trivial threefold moving in a fixed ambient space P. Suppose X contains a continuous family of curves, all of whose members satisfy certain unobstructedness conditions in P. A formula is given for computing the corresponding virtual number of curves, that is, the number of curves on a generic deformation of X "contributed by" the continuous family on X.
We investigate necessary conditions for Gorenstein projective varieties to admit semiorthogonal decompositions introduced by Kawamata, with main emphasis on threefolds with isolated compound $A_n$ singularities. We introduce obstructions coming from Algebraic $\mathrm{K}$-theory and translate them into the concept of maximal nonfactoriality. Using these obstructions we show that many classes of nodal threefolds do not admit Kawamata type semiorthogonal decompositions. These include nodal hypersurfaces and double solids, with the exception of a nodal quadric, and del Pezzo threefolds of degrees…
We study nodal complete intersection threefolds of type $(2,4)$ in $\PP^5$ which contain an Enriques surface in its Fano embedding. We completely determine Calabi-Yau birational models of a generic such threefold. These models have Hodge numbers $(h^{11},h^{12})=(2,32)$. We also describe Calabi-Yau varieties with Hodge numbers equal to $(2,26)$, $(23,5)$ and $(31,1)$. The last two pairs of Hodge numbers are, to the best of our knowledge, new.
We continue the development of methods for enumerating nodal curves on smooth complex surfaces, stressing the range of validity. We illustrate the new methods in three important examples. First, for up to eight nodes, we confirm Göttsche's conjecture about plane curves of low degree. Second, we justify Vainsencher's enumeration of irreducible six-nodal plane curves on a general quintic threefold in four-space. Third, we supplement Bryan and Leung's enumeration of nodal curves in a given homology class on an Abelian surface of Picard number one.
Chern-Ricci flat balanced metrics on small resolutions of Calabi-Yau threefolds
本文通过粘合构造证明，射影节点 Kähler Calabi-Yau 三流形的所有小分解均容许 Chern-Ricci 平衡度量，从而解决了 Hull-Strominger 系统的伸缩子方程。
We prove the following form of the Clemens conjecture in low degree. Let $d\le9$, and let $F$ be a general quintic threefold in $\IP^4$. Then (1)~the Hilbert scheme of rational, smooth and irreducible curves of degree $d$ on $F$ is finite, nonempty, and reduced; moreover, each curve is embedded in $F$ with normal bundle $Ø(-1)\oplusØ(-1)$, and in $\IP^4$ with maximal rank. (2)~On $F$, there are no rational, singular, reduced and irreducible curves of degree $d$, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3)~On $F$, there are no connected, reduced and reducible …
Categorical absorption of a non-isolated singularity
本文研究了非孤立奇点的三重曲面，通过构造半正交分解吸收奇点，并探讨其与节点品种的几何关系及平滑过程下的性质。
The boundary of K-moduli of prime Fano threefolds of genus twelve
本文研究了素数Fano三维流形 genus 十二的K-模空间，证明其边界由四个不可约分量组成，揭示了K-模空间与抗canonical K3表面的关系，为Mukai哲学提供了模表示。
Limits of nodal surfaces and applications
本文研究平坦族中δ-结点曲面趋于双重正规交叉中心纤维时的极限结构，证明极限曲面由两光滑曲面沿δ-结点曲线相交组成，并利用形变理论给出Severi簇正则不可约分支的存在性结果。
Constructive Proof of the Hodge Conjecture for K3 Surfaces via Nodal Degenerations
本文提出不依赖 Torelli 定理的复 K3 曲面 Hodge 猜想构造性证明，借助至多十个 A_1-节点的四次退化、Clemens–Schmid 序列与 Picard–Lefschetz 理论，将任意有理 (1,1)-类实现为代数闭链。
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