Related papers: Unexpected hypersurfaces with multiple fat points
The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in Algebraic Geometry. In general, it is still an open problem to understand when the points fail…
Our goal is to describe superconformal structures on super Riemann surfaces (SRS), based on data assigned to a fatgraph. We start from the complex structures on punctured $(1|1)$-supermanifolds, characterizing the corresponding moduli and…
Several uniqueness results on compact maximal hypersurfaces in a wide class of sta- bly causal spacetimes are given. They are obtained from the study of a distinguished function on the maximal hypersurface, under suitable natural first…
In the paper we provide a new method of proving the existence of a hypersurface of degree $d$ in $\mathbb{P}^n$, with a general point of multiplicity $m$ and vanishing at a given set of points $Z$, by looking at weak combinatorics of a set…
Exceptional points in non-Hermitian systems have recently been shown to possess nontrivial topological properties, and to give rise to many exotic physical phenomena. However, most studies thus far have focused on isolated exceptional…
We study hypersurfaces with fractional mean curvature in N-dimensional Euclidean space. These hypersurfaces are critical points of the fractional perimeter under a volume constraint. We use local inversion arguments to prove existence of…
We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal…
The purpose of this note is to establish a direct link between the theory of unexpected hypersurfaces and varieties with defective osculating behavior. We identify unexpected plane curves of degree 4 as sections of a rational surface X of…
We study negative curves on surfaces obtained by blowing up special configurations of points in the complex projective palne. Our main results concern the following configurations: very general points on a cubic, 3-torsion points on an…
We study the postulation of 0-dimensional schemes given by unions of 2-superfat points in general position in the plane, i.e., the union of local schemes defined by the intersection of two distinct double lines. We prove that such schemes…
A recent paper by the first and third authors together with Sabourin raised the question of what the possible Hilbert functions are for fat point subschemes of the form $2p_1+...+2p_r$, for all possible choices of $r$ distinct points in the…
We give an introduction to the study of algebraic hypersurfaces, focusing on the problem of when two hypersurfaces are isomorphic or close to being isomorphic. Working with hypersurfaces and emphasizing examples makes it possible to discuss…
The relationship between interpolation and separation properties of hypersurfaces in Bargmann-Fock spaces over $\mathbb{C} ^n$ is not well-understood except for $n=1$. We present four examples of smooth affine algebraic hypersurfaces that…
In connection with his counter-example to the fourteenth problem of Hilbert, Nagata formulated a conjecture concerning the postulation of r fat points of the same multiplicity in the projective plane and proved it when r is a square.…
In this work we study some problems related with algebraic hypersurfaces invariant by foliations on weighted projective spaces $\mathbb{P}_{\mathbb{C}}(\varpi_0,...,\varpi_n)$ generalizing some results known for $\p$, as for example: the…
Traditional algebraic geometric invariants lose some of their potency in positive characteristic. For instance, smooth projective hypersurfaces may be covered by lines despite being of arbitrarily high degree. The purpose of this…
An expression for the first variation of the area functional of the second fundamental form is given for a hypersurface in a semi-Riemannian space. The concept of the "mean curvature of the second fundamental form" is then introduced. Some…
In this paper, we study hypersurfaces of Euclidean spaces with arbitrary dimension. First, we obtain some results on $\mbox{H}$-hypersurfaces. Then, we give the complete classification of $\mbox{H}$-hypersurfaces with 3 distinct curvatures.…
We study the bi-graded Hilbert function of ideals of general fat points with same multiplicity in $\mathbb{P}^1\times\mathbb{P}^1$. Our first tool is the multiprojective-affine-projective method introduced by the second author in previous…
In (equi-)affine differential geometry, the most important algebraic invariants are the affine (Blaschke) metric h, the affine shape operator S and the difference tensor K. A hypersurface is said to admit a pointwise symmetry if at every…