A remark on K3s of Todorov type (0,9) and (0,10)

Remarks on hard Lefschetz conjectures on Chow groups

We propose two conjectures of Hard Lefschetz type on Chow groups and prove them for some special cases. For abelian varieties, we shall show they are equivalent to well-known conjectures of Beauville and Murre.

A remark on the Chow ring of Sicilian surfaces

We propose a "Bloch type" conjecture for surfaces: if the cup product map in coherent cohomology is zero, then all intersections of homologically trivial divisors should be zero in the Chow group of zero-cycles. We prove this conjecture for Sicilian surfaces.

关于 K3 型 Brauer 类的一些注记

Some remarks on Brauer Classes of K3-type

研究了复射影 K3 曲面 Brauer 群中素数阶元素定义的模空间有限映射，计算了其次数，并揭示了 Picard 数为 2 时 Picard 格的决定关系。

Remarks on Type III Unprojection

Type III unprojection plays a very important role in the birational geometry of Fano threefolds. It was first introduced by A. Corti on his calculations of Fano threefolds of genus 6 and 7. It seems that at present a general definition of the type III unprojection is still missing. After proving some general facts about residual ideals, we propose a definition for the generic Type III unprojection, and prove that it gives a Gorenstein ring.

A remark on periodic points on varieties over a field of finite type over Q

Let M be a field of finite type over {\bf Q} and X a variety defined over M. We study when the set {P \in X(K) \mid f^{\circ n} (P) = P for some n \geq 1} is finite for any finite extension fields K of M and for any dominant K-morphisms f : X \to X with deg f \geq 2.

A remark on a Nagell-Lutz type statement for the Jacobian of a curve of genus 2 and a (2, 3, 6) quasi-torus decomposition of a sextic with 9 cusps

In this note, an analogous statement to the Nagell-Lutz theorem does not hold for the Jacobian of a certain curve of genus 2 over $\mathbb{C}(t)$. As a by-product, we give a (2, 3, 6) quasi-torus decomposition for the dual curve of a smooth cubic.

A remark on potentially semi-stable representations of Hodge-Tate type (0,1)

In this note we complement a part of a theorem of Fontaine-Mazur. We show that if $(V,ρ)$ is an irreducible finite dimensional representation of the Galois group $Gal({\bar K}/K)$ of a finite extension of $K\Q_p$, of Hodge-Tate type $(0,1)$ then it is potentially semi-stable if and only if it is potentially crystalline. This was proved by Fontaine-Mazur for dimension two and $p\geq 5$ by their classfication theorem.

Remarks on level one conformal blocks divisors

We show that conformal blocks divisors of type B_r and D_r at level one are effective sums of boundary divisors of $\bar{M}_{0,n}$. We also prove that the conformal blocks divisor of type $B_r$ at level 1 with weights (ω_1,\dots,ω_1) scales linearly with the level.

一个博戈莫洛夫型消失定理

A remark on Bogomolov type vanishing theorem

研究在射影流形上建立博戈莫洛夫型消失定理，核心方法是基于数值有效线丛和乘子理想层，给出相关定理。