Liouville定理的一个类比及其在三次曲面上的应用

四次三次折叠中直线曲面的Chow群

Chow groups of surfaces of lines in cubic fourfolds

研究四次三次折叠中与固定直线相交的直线曲面的动机分解，定义Beauville-Voisin类的类似物，并利用Shen和Vial引入的Bloch-Beilinson滤过自然分裂研究到所有直线Fano簇的前推映射。

四次三维簇的Fano簇上的常循环曲面

Constant cycle surfaces on Fano varieties of cubic fourfolds

本文研究光滑四次三维簇的线Fano簇上常循环曲面的有限性，证明了固定阶的形如F(Y)的常循环曲面至多有限个。

关于三次曲面的Mordell-Weil生成元个数

On the number of Mordell-Weil generators for cubic surfaces

设S是域K上的光滑三次曲面。众所周知，新的K-有理点可以通过割线和切线构造从旧的K-有理点获得。Mordell-Weil生成集是S（K）的最小基数子集B，它通过连续的割线和切线构造生成S（K）。设r（S，K）是这样一个Mordell-Weil生成集的基数。马宁提出了被称为三次曲面的莫德尔-韦伊问题：如果K是在其素子域上生成的，那么r（S，K）是有限的。本文证明了这一猜想的一个特例。也就是说，如果S包含两条定义在K上的斜直线，则r（S，K）= 1。研究三次曲面上的割线和切线过程的困难之一是它不像椭圆曲线那样导致结合的二元运算。作为一种部分的补救，我们引入了一个阿贝尔群H_S（K），它与一个由K-有理点自然生成的三次曲面S/K相联系，它保留了许多关于割线和切线过程的信息。特别地，只要H_S（K）的最小生成元数大，则r（S，K）也大。在弱近似成立的情况下，H_S具有良好的局部到全局性质。我们利用这些构造一族有理数上的光滑三次曲面，使得r（S，K）在这个族中是无界的.这是椭圆曲线秩猜想无界性的三次曲面模拟。

Ranks and Symmetric Ranks of Cubic Surfaces

We study cubic surfaces as symmetric tensors of format $4 \times 4 \times 4$. We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The corresponding algebraic problem concerns border ranks. We show that the non-symmetric border rank coincides with the symmetric border rank for cubic surfaces. As part of our analysis, we obtain minimal ideal generators for the symmetric analogue to the secant variety from the salmon conjecture. We also give a test for symmetric rank given by the non-vanishing …

Nodal quintic surfaces and lines on cubic fourfolds

We study nodal quintic surfaces with an even set of 16 nodes as analogues of singular Kummer surfaces. The interpretation of the natural double cover of an even 16-nodal quintic as a certain Fano variety of lines could be viewed as a replacement for the additive structure of the cover of a singular Kummer surface by its associated abelian surface. Most of the results in this article can be seen as refinements of known facts and our arguments rely heavily on techniques developed by Beauville, Murre, and Voisin. Results due to Izadi and Shen are particularly close to some of the statements. In t…