The tangential ramification locus $B_{X/Y}^t\subset B_{X/Y}$ is the subset of points in the ramification locus where the sheaf of relative vector fields $T_{X/Y}$ fails to be locally free. It was conjectured by Zariski and Lipman that if $V/k$ is a variety over a field $k$ of characteristic 0 and $B^t_{V/k}= \emptyset$, then $V/k$ is smooth (=regular). We prove this conjecture when $V/k$ is a locally complete intersection. We prove also that $B_{V/k}^t= \emptyset$ implies $\operatorname {codim}_V B_{V/k}\leq 1 $ in positive characteristic, if $V/k $ is the fibre of a flat morphism satisfying generic smoothness.
We study the failure of the Lipman-Zariski conjecture in positive characteristic. For rational double points, the conjecture holds true except for a short finite list of exceptions. For log canonical surface singularities, the conjecture continues to hold with the same list of exceptions under an additional tameness hypothesis. In particular, among rational double points in characteristic $p \ge 7$ Lipman's counterexample is the only one, and the conjecture holds for all tame $F$-pure normal surface singularities.
We prove the Lipman-Zariski conjecture for complex surface singularities of genus one, and also for those of genus two whose link is not a rational homology sphere. As an application, we characterize complex $2$-tori as the only normal compact complex surfaces whose smooth locus has trivial tangent bundle. We also deduce that all complex-projective surfaces with locally free and generically nef tangent sheaf are smooth, and we classify them.
利用$p_g - g - b \le 2$证明了复曲面奇性的Lipman-Zerkki猜想。这里$p_g$是几何亏格，$g$是例外曲线的亏格之和，$b$是对偶图的第一个贝蒂数.这改进了第二作者以前的结果。作为应用，我们证明了具有局部自由切层的紧致复曲面是光滑的，只要它允许两个一般线性无关的扭曲向量场，并且它的标准层至多有两个整体瓣.
Let $E$ be a smooth cubic. A plane curve $D$ is said to be an $n$-contact curve to $E$ if the intersection multiplicities at each intersection point between $E$ and $D$ is $n$. In this note, we give an algorithm to produce $n$-contact curves to $E$ and consider its application.
We prove that any nef b-divisor class on a projective variety defined over an algebraically closed field of characteristic 0 is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef b-divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony. We show that any big and basepoint free curve class is a power of a nef b-divisor, and relate this statement to Zariski decompositions of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces …
The Complete Intersection Discrepancy of a Curve II: Families of Curves
Institute of Mathematics and Informatics Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Department of Mathematics Northeastern University（东北大学数学系）；Institut de Mathématiques de Jussieu-Paris Rive Gauche Centre national de la recherche scientifique (CNRS)（巴黎左岸苏邦数学研究所 法国国家科学研究中心）
