Alexander Samokhin
In this paper we prove a cohomology vanishing result for quadrics of low dimension in positive characteristic. This vanishing is a necessary condition for the D-affinity of these varieties.
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We compute decomposition of Frobenius push-forwards of line bundles on quadrics into a direct sum of line bundles and spinor bundles. As an application we show when the Frobenius push-forward gives a tilting bundle and we apply it to study D-modules on quadrics.
$D$-affinity of Quadrics Revisited
本文重新探讨了二次曲面的$D$-affinity性质，证明在特定条件下二次超曲面和格拉斯曼流形不是$D$-affine，并补充了之前关于$D$-affine条件的研究。
Let $G$ be a simple algebraic group of type $B_2$ over an algebraically closed field of odd characteristic. We prove that the flag variety $G/B$ is D-affine. This extends an earlier result of H.H.Andersen and M.Kaneda.
Comparing vanishing of local cohomology in zero and positive characteristic, we give an example of a D-module on a Grassmann variety in positive characteristic with non-vanishing first cohomology group. This is a counterexample to D-affinity and the Beilinson-Bernstein equivalence for flag manifolds in positive characteristic.
This article is a continuation of a previous article which concerned the splitting problem for subspaces of superspaces. We begin with a general account of projective superspaces. Subsequently, we specialise to subvarieties of `positive' projective superspaces. Our main result is: positive, projective superspaces are `normal', in a sense we define. Then, among others, our main application is: smooth, non-reduced, superspace quadric hypersurfaces are non-split.
We classify all positive integers n and r such that (stably) non-rational complex r-fold quadric bundles over rational n-folds exist. We show in particular that for any n and r, a wide class of smooth r-fold quadric bundles over projective n-space are not stably rational if r lies in the interval from $2^{n-1}-1$ to $2^{n}-2$. In our proofs we introduce a generalization of the specialization method of Voisin and Colliot-Thélène--Pirutka which avoids universally $CH_0$-trivial resolutions of singularities.
Let $C$ be a smooth curve of genus $g$. For each positive integer $r$ the $r$-gonality $d_r(C)$ of $C$ is the minimal integer $t$ such that there is $L\in {Pic}^t(C)$ with $h^0(C,L) =r+1$. In this paper for all $g\ge 40805$ we construct several examples of smooth curves $C$ of genus $g$ with $d_3(C)/3< d_4(C)/4$, i.e. for which a slope inequality fails.
For even dimensional smooth complete intersections, of dimension at least 4, of two quadric hypersurfaces in a projective space, we study the genus zero Gromov-Witten invariants by the monodromy group of its whole family. We compute the invariants of length 4 and show that, besides a special invariant, all genus zero Gromov-Witten invariants can be reconstructed from the invariants of length 4. In dimension 4, we compute the special invariant by solving a curve counting problem. We show that the generating function of genus zero Gromov-Witten invariants has a positive radius of convergence. We…
Fano threefolds in positive characteristic I
本文在正特征下研究了 Picard 数为一的 Fano 三维流形，证明了高 genus 的 Fano 流形是二次曲面的交集。
Let $X$ be a smooth variety over an algebraically closed field $k$ of positive characteristic, ${\rm D}_X$ the sheaf of PD-differential operators, and ${\bar D}_X$ its central reduction, the sheaf of small differential operators. In this paper we show that if $X$ is a line-hyperplane incidence variety (a partial flag variety of type $(1,n,n+1)$) or a quadric of arbitrary dimension (in this case the characteristic is supposed to be odd) then ${\rm H}^{i}(X,{\bar D}_X)=0$ for $i>0$. Using this vanishing result and the derived localization theorem for crystalline differential operators (\cite{BMR…
On The Cohomology of $N_C(-2)$ in Positive Characteristic
本文将一般 Brill--Noether 曲线二次截面控制条件 $H^0(N_C(-2))=0$ 的经典结果从特征零推广到正特征，并发现特征 2 下存在大量新例外。
Generic Triviality of Linear Automorphisms of Complete Intersections
本文证明在代数闭域上，正维数且余维数至少为2的一般光滑完全交的线性自同构群平凡，仅两个二次曲面例外，证明基于关联簇并区分处理两类自同构。
We construct the moduli space of Enriques surfaces in positive characteristic and eventually over the integers, and determine its local and global structure. As an application, we show lifting of Enriques surfaces to characteristic zero. The key observation is that the canonical double cover of an Enriques surface is birational to the complete intersection of three quadrics in IP^5, even in characteristic 2.
The rationality problem for hypersurfaces of degree at least five
本文证明次数至少为五的非常一般超曲面不具有对角分解，从而非稳定或收缩有理，通过构造具有非零非分歧上同调类且由二次曲面纤维化的五次超曲面完成证明。
Characteristic-free characterizations of projective space for smooth toric varieties
该研究证明光滑射影环簇切丛含正秩丰富局部自由子层时同构于射影空间，还建立了Beauville上同调刻画的环簇形式，且指出无环簇假设时相关刻画在特征2下不成立。
Integers expressible as the sum of two rational cubes
本文证明了正整数比例的整数可表示为两个有理数立方之和，并推广证明三次扭曲族中至少六分之一的椭圆曲线秩为0或1，核心方法是结合二元三次型参数化、几何数论与圆法计算2-Selmer群平均大小。
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