Related papers: On higher Hessians and the Lefschetz properties

Given an embedded smooth projective variety Y in CP^n, we show how the existence of a hypersurface with high multiplicity along Y, but of relatively low degree and log canonical near Y implies vanishing of higher cohomology for certain…

Once first answers in any dimension to the Green-Griffiths and Kobayashi conjectures for generic algebraic hypersurfaces $\mathbb{X}^{n-1} \subset \mathbb{P}^n(\mathbb{C})$ have been reached, the principal goal is to decrease (to improve)…

Let Z be a general surface in P^3 of degree at least 5. Using a Lefschetz pencil argument, we give an elementary new proof of the vanishing of a regulator on K_1(Z).

We compute the essential dimension of the functors Forms_{n,d} and Hypersurf_{n, d} of equivalence classes of homogeneous polynomials in n variables and hypersurfaces in P^{n-1}, respectively, over any base field k of characteristic 0. Here…

Recently, the authors obtained the Schur multiplier, the non-abelian tensor square and the non-abelian exterior square of $d$-generator generalized Heisenberg Lie algebras of rank $ \frac{1}{2}d(d-1).$ Here, we intend to obtain the same…

For an affine toric variety $\spec(A)$, we give a convex geometric interpretation of the Gerstenhaber product $\HH^2(A)\times \HH^2(A)\to \HH^3(A)$ between the Hochschild cohomology groups. In the case of Gorenstein toric surfaces we prove…

Let X be an irreducible hypersurface in $\mathbb{P}^n$ of degree $d\geq 3$ with only isolated semi-weighted homogeneous singularities, such that $exp(\frac{2\pi i}{k})$ is a zero of the Alexander polynomial. Then we show that the…

This note is an appendix to 'Measures of irrationality for hypersurfaces of large degree' by L. Ein, R. Lazarsfeld and B. Ullery. We prove an existence result for families of curves having low gonality, and lying on fundamental loci of…

In the first part of this paper, we establish some results around generalized Borel's Theorem. As an application, in the second part, we construct example of smooth surface of degree $d\geq 19$ in $\mathbb{CP}^3$ whose complements is…

We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree $d$, over any global field. In particular, we focus on the affine hypersurface situation by…

Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in $\mathbb{P}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot…

The paper is devoted to the investigation of finite dimensional commutative nilpotent (associative) algebras N over an arbitrary base field of characteristic zero. Due to the lack of a general structure theory for algebras of this type (as…

We discuss a paper of M. Green from a new algebraic perspective, and provide applications of its results to level and Gorenstein algebras, concerning their Hilbert functions and the weak Lefschetz property. In particular, we will determine…

We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset \mathbb P^{n+1}$ of degree $d$ over an algebraically closed…

We provide a classification of developable cubic hypersurfaces in $\mathbb P^4$. Using the correspondence between forms of degree $3$ on $\mathbb P^4$ and Artinian Gorenstein $\mathbb K$-algebras, given by Macaulay-Matlis duality, we…

Let A be a standard graded Artinian algebra over a field of characteristic zero and let z be a linear form in A. We define the central simple modules for each such pair (A, z). Assume that A is Gorenstein. Then we prove that A has the…

Wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank. The discrepancy between these two ranks is caused by the difference between the limit of spans of a family of zero-dimensional schemes…

This paper initiates a systematic study for key properties of Artinian Gorenstein \(K\)-algebras having binomial Macaulay dual generators. In codimension 3, we demonstrate that all such algebras satisfy the strong Lefschetz property, can be…

The Noether-Lefschetz theorem asserts that any curve in a very general surface $X$ in $\mathbb P^3$ of degree $d \geq 4$ is a restriction of a surface in the ambient space, that is, the Picard number of $X$ is $1$. We proved previously that…

We study the categorical Torelli theorem for smooth (weighted) hypersurfaces in (weighted) projective spaces via the Hochschild--Serre algebra of its Kuznetsov component. In the first part of the paper, we show that a natural graded…