Some observations on local and projective hypersurfaces

Hassett-Tschinkel correspondence: Modality and projective hypersurfaces

B. Hassett and Yu. Tschinkel (1999) introduced a remarkable correspondence between generically transitive actions of a commutative unipotent algebraic group G and finite-dimensional local algebras. In this paper we develop Hassett-Tschinkel correspondence and calculate modality of generically transitive G-actions on projective spaces, classify actions of modality one, and characterize generically transitive G-actions on projective hypersurfaces of given degree. In particular, actions on degenerate projective quadrics are studied.

射影超曲面的拟齐次奇点与Jacobian合冲

Quasi-homogeneous singularities of projective hypersurfaces and Jacobian syzygies

该研究发现射影超曲面Jacobian合冲与奇点性质的一般关系，提出识别拟齐次孤立奇点的新方法，推广了此前关于自由与殆自由平面曲线的研究成果。

Projective Duality, Unexpected Hypersurfaces and Logarithmic Derivations of Hyperplane Arrangements

Several papers have been written studying unexpected hypersurfaces. We say a finite set of points Z admits unexpected hypersurfaces if a general union of fat linear subspaces imposes less that the expected number of conditions on the ideal of Z. In this paper, we introduce the concept of a very unexpected hypersurface. This is a stronger condition which takes into account an explanation for some hypersurfaces previously considered unexpected. We then develop a duality theory to relate the study of very unexpected hypersurfaces to the derivations of dual hyperplane arrangements. This allows us …

Jacobian syzygies, stable reflexive sheaves, and Torelli properties for projective hypersurfaces with isolated singularities

We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a projective hypersurface $V$ with isolated singularities and the Torelli properties of $V$ (in the sense of Dolgachev-Kapranov). We show in particular that hypersurfaces with a small Tjurina numbers are Torelli in this sense. When $V$ is a plane curve, or more interestingly, a surface in $P^3$, we discuss the stability of the reflexive sheaf of logarithmic vector fields along $V$. A new lower bound for the minimal degree of a syzygy associated to a 1-dimensional complete intersection is also g…

关于曲线和Frobenius分裂超曲面的行列式方程

On determinantal equations for curves and Frobenius split hypersurfaces

本文研究了射影空间中曲线和超曲面的内在行列式方程存在性问题，证明了在多种正特征情况下平面射影曲线和Frobenius分裂超曲面均存在内在行列式或Pfaffian表示。

Hypersurfaces with linear type singular loci

In this paper, necessary and sufficient criteria for the Jacobian ideal of a reduced hypersurface with isolated singularity to be of linear type, are presented. We prove that the gradient ideal of a reduced projective plane curve with simple singularities ($\mathrm{ADE}$) is of linear type. We show that any reduced projective quartic curve is of gradient linear type.