Related papers: Homaloidal hypersurfaces and hypersurfaces with va…

If $X = V(f) \subset \mathbb P^N$ is a reduced complex hypersurface, the hessian of $f$ (or by abusing the terminology the hessian of $X$) is the determinant of the matrix of the second derivatives of the form $f$, that is the determinant…

Hypersurfaces are studied and classified under multiple additional assumptions in any Riemannian homogeneous space $(\mathbb{C}P^3, g_a)$, including nearly K\"ahler $\mathbb{C}P^3$. Notably, all extrinsically homogeneous hypersurfaces are…

We deal with a generalization of a Theorem of P. Gordan and M. Noether on hypersurfaces with vanishing (first) Hessian. We prove that for any given $N\geq 3$, $d \geq 3$ and $2\leq k < \frac{d}{2}$ there are infinitely many irreducible…

A (global) determinantal representation of hypersurface in P^n is a matrix, whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface. We study the properties of such representations for…

We present a general construction of hypersurfaces with vanishing hessian, starting from any irreducible non-degenerate variety whose dual variety is a hypersurface and based on the so called Dual Cayley Trick. The geometrical properties of…

In this article we extend the notion of determinantal representation of hypersurfaces to the determinantal representation of sections of the determinant line bundle of a vector bundle. We give several examples, and prove some necessary…

Hesse claimed that an irreducible projective hypersurface in $\PP^n$ defined by an equation with vanishing hessian determinant is necessarily a cone. Gordan and Noether proved that this is true for $n\leq 3$ and constructed counterexamples…

We prove the existence of rotational hypersurfaces in $\mathbb{H}^n\times \mathbb{R}$ with $H_{r+1}=0$ and we classify them. Then we prove some uniqueness theorems for $r$-minimal hypersurfaces with a given (finite or asymptotic) boundary.…

We explore the existence of irreducible and reducible arc-sections in an irreducible hypersurface singularity germ along finite projections. In particular we provide examples of irreducible isolated hypersurface singularities for which no…

Given a smooth quasi-projective complex algebraic variety $\mathcal{S}$, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over $\mathcal{S}$ of degree $d$ in…

We discuss families of hypersurfaces with isolated singularities in projective space with the property that the sum of the ranks of the rational homotopy and the homology groups is finite. They represent infinitely many distinct homotopy…

We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining a nodal hypersurface. The result gives information on the position of the singularities of a nodal hypersurface…

For a subvariety of a smooth projective variety, consider the family of smooth hypersurfaces of sufficiently large degree containing it, and take the quotient of the middle cohomology of the hypersurfaces by the cohomology of the ambient…

We show that general moving enough families of high enough degree hypersurfaces in a complex projective space do not have a dominant set of sections.

We wish to attack the problems that H.~Anciaux and K.~Panagiotidou posed in [1], for non-degenerate real hypersurfaces in indefinite complex projective space. We will slightly change these authors' point of view, obtaining cleaner equations…

Let $\mathscr{X}\to W$ be a flat family of generically irreducible hypersurfaces of degree $d\geq 2$ in $\PP^n$ with singular locus of dimension $t$, with $W$ unirational of dimension $r$. We prove that if $n$ is large enough with respect…

We study the problem of the irreducibility of the Hessian variety $\mathcal{H}_f$ associated with a smooth cubic hypersurface $V(f)\subset \mathbb{P}^n$. We prove that when $n\leq5$, $\mathcal{H}_f$ is normal and irreducible if and only if…

We prove that a general determinantal hypersurface of dimension 3 is nodal. Moreover, in terms of Chern classes associated with bundle morphisms, we derive a formula for the intersection homology Euler characteristic of a general…

Equivalences under the affine group ${\rm Aff} (\mathbb{R}^3)$ of constant Hessian rank $1$ surfaces $S^2 \subset \mathbb{R}^3$, sometimes called parabolic, were, among other objects, studied by Doubrov, Komrakov, Rabinovich, Eastwood,…

We determine all affinely homogeneous hypersurfaces S^3 in R^4 whose Hessian is (invariantly) of constant rank 2, including the simply transitive ones. We find 34 inequivalent terminal branches yielding each to a nonempty moduli space of…