Boundary manifolds of projective hypersurfaces

Cohomology rings and algebraic torus actions on hypersurfaces in the product of projective spaces and bounded flag varieties

In this paper, for any Milnor hypersurface we find the largest dimension of effective algebraic torus actions on it. The proof of the corresponding theorem is based on the computation of the automorphism group for any Milnor hypersurface. We find all generalised Buchstaber-Ray and Ray hypersurfaces that are toric varieties. We compute the Betti numbers of these hypersurfaces and describe their integral singular cohomology rings in terms of the cohomology of the corresponding ambient varieties.

关于扫描映射与光滑射影超曲面空间

On the scanning map and the space of smooth projective hypersurfaces

本文研究CPⁿ中d次光滑射影超曲面空间的扫描映射，通过计算U(n+1)共轭作用的有理同伦模型，得出d>2时扫描映射在同调上诱导单态射的结论。

Computing Milnor fiber monodromy for some projective hypersurfaces

We describe an algorithm computing the monodromy and the pole order filtration on the top Milnor fiber cohomology of hypersurfaces in $\mathbb{P}^n$ whose pole order spectral sequence degenerates at the second page. In the case of hyperplane arrangements and free, locally quasi-homogeneous hypersurfaces, and assuming a key conjecture, this algorithm is much faster than for a hypersurface as above. Our conjecture is supported by the results due to L. Narv\' ez Macarro and M. Saito on the roots of Bernstein-Sato polynomials of such hypersurfaces, by all the examples computed so far, and by one p…

加权射影空间超曲面的上同调与 $H^2$ 上的相交形式

Cohomology of hypersurfaces of weighted projective space and the intersection form on $H^2$

本文通过计算加权射影空间超曲面的第二上同调相交形式并描述低次整系数上同调群，为识别奇异 Fano 簇光滑化得到的 Fano 流形提供了显式工具。

Representations on the cohomology of smooth projective hypersurfaces with symmetries

This paper is concerned with the primitive cohomology of a smooth projective hypersurface considered as a linear representation for its automorphism group. Using the Lefschetz-Riemann-Roch formula, the character of this representation is described on each piece of the Hodge decomposition. A consequence about the existence of smooth symmetric hypersurfaces that are stable under the standard irreducible permutation representation of the symmetric group on homogeneous coordinates is drawn.

Average size of the automorphism group of smooth projective hypersurfaces over finite fields

In this paper, we show that the average size of the automorphism group over $\mathbb{F}_q$ of a smooth degree $d$ hypersurface in $\mathbb{P}^{n}_{\mathbb{F}_q}$ is equal to $1$ as $d\rightarrow \infty$. We also discuss some consequence of this result for the moduli space of smooth degree $d$ hypersurfaces in $\mathbb{P}^n$.