On a conjecture of Tsfasman and an inequality of Serre for the number of points on hypersurfaces over finite fields

Complete Description of Measures Corresponding to Abelian Varieties over Finite Fields

We study probability measures corresponding to families of abelian varieties over a finite field. These measures play an important role in the Tsfasman- Vladuts theory of asymptotic zeta-functions defining completely the limit zeta-function of the family. J.-P.Serre, using results of R.M.Robinson on conjugate algebraic integers, described the possible set of measures than can correspond to families of abelian varieties over a finite field. The problem whether all such measures actually occur was left open. Moreover, Serre supposed that not all such measures correspond to abelian varieties (for…

加权射影空间上有限域上超曲面的有理点最大数

Maximum number of rational points on hypersurfaces in weighted projective spaces over finite fields

本文在第二重量为1的条件下，证明了加权射影空间上有限域中超曲面有理点最大数的猜想，扩展了之前在二维情况下的结果。

Serre代数、矩阵分解与超曲面的范畴Torelli定理

Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces

本文为Fano簇的Kuznetsov分量构造Serre代数，对Fano超曲面建立其子代数与Jacobi环的联系，并由此证明一类Fano超曲面的范畴Torelli定理。

Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory

We consider the question of determining the maximum number of $\mathbb{F}_q$-rational points that can lie on a hypersurface of a given degree in a weighted projective space over the finite field $\mathbb{F}_q$, or in other words, the maximum number of zeros that a weighted homogeneous polynomial of a given degree can have in the corresponding weighted projective space over $\mathbb{F}_q$. In the case of classical projective spaces, this question has been answered by J.-P. Serre. In the case of weighted projective spaces, we give some conjectures and partial results. Applications to coding theo…

通过Hochschild-Serre代数对Kuznetsov Fano三维簇猜想的否定

Kuznetsov's Fano threefold conjecture via Hochschild-Serre algebra

本文通过矩阵分解研究光滑四次双重立体的Kuznetsov分量的Hochschild-Serre代数乘法，并以此给出Kuznetsov Fano三维簇猜想的新反证。

Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields

We consider the problem of determining the maximum number of common zeros in a projective space over a finite field for a system of linearly independent multivariate homogeneous polynomials defined over that field. There is an elaborate conjecture of Tsfasman and Boguslavsky that predicts the maximum value when the homogeneous polynomials have the same degree that is not too large in comparison to the size of the finite field. We show that this conjecture holds in the affirmative if the number of polynomials does not exceed the total number of variables. This extends the results of Serre (1991…