高阶对角三次曲面的Tamagawa数

On the universal $\mathrm{CH}_0$ group of cubic threefolds in positive characteristic

We adapt for algebraically closed fields $k$ of characteristic greater than $2$ two results of Voisin, on the decomposition of the diagonal of a smooth cubic hypersurface $X$ of dimension $3$ over $\mathbb C$, namely: the equivalence between Chow-theoretic and cohomological decompositions of the diagonal of those hypersurfaces and the fact that the algebraicity (with $\mathbb Z_2$-coefficients) of the minimal class $θ^4/4!$ of the intermediate jacobian $J(X)$ of $X$ implies the Chow-theoretic decomposition of the diagonal of $X$. Using the second result, the Tate conjecture for divisors on sur…

On the universal $CH_0$ group of cubic hypersurfaces

We study the existence of a Chow-theoretic decomposition of the diagonal of a smooth cubic hypersurface, or equivalently, the universal triviality of its ${\rm CH}_0$-group. We prove that for odd dimensional cubic hypersurfaces or for cubic fourfolds, this is equivalent to the existence of a cohomological decomposition of the diagonal, and we translate geometrically this last condition. For cubic threefolds $X$, this turns out to be equivalent to the algebraicity of the minimal class $θ^4/4!$ of the intermediate Jacobian $J(X)$. In dimension $4$, we show that a special cubic fourfold with disc…

特殊三次零点与对偶簇

Special cubic zeros and the dual variety

本文基于对偶簇和 delta 方法，无条件提取了 $6$ 变量对角三次型特殊整零点的加权和，并提出了适用于更一般情况的公理化框架。

四次超曲面对角线和(2,3)-完全交集n-折叠的对角线分解

On the diagonal of quartic hypersurfaces and $(2,3)$-complete intersection $n$-folds

研究非常一般的四次超曲面和(2,3)-完全交集n-折叠的对角线分解问题，利用循环理论技术与Voisin的结果，扩展了稳定有理性到对角线分解存在的结论，并给出新证明。

On the Brauer group of affine diagonal quadrics

In a previous work, we introduced the notion of uniform generators of the Brauer group and proved that general diagonal cubic surfaces do not have such generators. In this paper, we prove that a similar non-existence result holds for affine diagonal quadrics.

Prelog Chow groups of self-products of degenerations of cubic threefolds

It is unknown whether smooth cubic threefolds have an (integral Chow-theoretic) decomposition of the diagonal, or whether they are stably rational or not in general. As a first step towards making progress on these questions, we compute the (saturated numerical) prelog Chow group of the self-product of a certain degeneration of cubic threefolds.

Koszul property of projections of the Veronese cubic surface

Let V be the Veronese cubic surface in P^9. We classify the projections of V to P^8 whose coordinate rings are Koszul. In particular we obtain a purely theoretical proof of the Koszulness of the pinched Veronese, a result obtained originally by Caviglia using filtrations, deformations and computer assisted computations. To this purpose we extend, to certain complete intersections, results of Conca, Herzog, Trung and Valla concerning homological properties of diagonal algebras.

On the Chow groups of the variety of lines of a cubic fourfold

Let $X$ be a smooth complex cubic fourfold and let $F$ be the variety of lines of $X$. The variety $F$ is known to be a smooth projective hyperkaehler fourfold, which is moreover endowed with a self rational map $ϕ: F -\rightarrow F$ first constructed by C. Voisin. Here we define a filtration of Bloch--Beilinson type on the Chow group of zero-cycles $CH_0(F)$ which canonically splits under the action of $ϕ$, thereby answering in this case a question of A. Beauville. Moreover, we show that this filtration is of motivic origin, in the sense that it arises from a Chow--Kuenneth decomposition of t…

$R$-等价于立方面I：已有非平凡普遍等价的情况

$R$-equivalence on Cubic Surfaces I: Existing Cases with Non-Trivial Universal Equivalence

本文研究了具有非平凡普遍等价的立方面，发现2-进数立方面在特定条件下$R$-等价为平凡或指数2，并验证了特定例子的平凡性，同时探讨了生成AI在研究中的应用。

三次曲面的行列式表示的模空间与根系的不变量理论

The Moduli Space of Determinantal Representations of Cubic Surfaces and Invariant Theory of Root Systems

该研究构建框架研究复射影三次曲面平展族的线性行列式表示模空间，针对E₆型有理二重点与Ẽ₆型简单椭圆奇点两种情形，分别建立相关同构结果，规避了局部单值群无限带来的框架失效问题。