Related papers: Dessins d'Enfants and Hypersurfaces with Many $A_j…
A long standing question is if maximum number $\mu(d)$ of nodes on a surface of degree $d$ in $\dP^3(\dC)$ can be achieved by a surface defined over the reals which has only real singularities. The currently best known asymptotic lower…
It is classically known that a real cubic surface in the real projective 3-space cannot have more than one solitary point (locally given by x^2+y^2+z^2=0) whereas it can have up to four nodes (x^2+y^2-z^2=0). We show that on any surface of…
Recently, W. Barth and S. Rams discussed sextics with up to 30 $A_2$-singularities (also called cusps) and their connection to coding theory [math.AG/0403018]. In the present paper, we find a sextic with 35 cusps within a four-parameter…
We prove the factoriality of a nodal hypersurface in $\mathbb{P}^{4}$ of degree $d$ that has at most $2(d-1)^{2}/3$ singular points, and factoriality of a double cover of $\mathbb{P}^{3}$ branched over a nodal surface of degree $2r$ having…
In this note we investigate three new pencils of symmetric surfaces in complex projective three-space. These have degree 6, 8 resp. 12 and are invariant under the action of subgroups of SO(4) containing the Heisenberg group. The pencils of…
In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double…
Using degeneration to scrolls, we give an easy proof of non-existence of curves of low genera on general surfaces in P3 of degree d >=5. We show, along the same lines, boundedness of families of curves of small enough genera on general…
It is well known since Noether that the gonality of a smooth plane curve of degree d>3 is d-1. Given a k-dimensional complex projective variety X, the most natural extension of gonality is probably the degree of irrationality, that is the…
Until now, the only known maximal surfaces in Minkowski 3-space of finite topology with compact singular set and without branch points were either genus zero or genus one, or came from a correspondence with minimal surfaces in Euclidean…
We study equisingular deformation problems for curves and surfaces in algebraic families, with particular emphasis on situations where nodal behavior is no longer generic. Extending classical Severi theory, we develop deformation--theoretic…
In this paper the singular hypersurfaces in $\mathbb{C}\mathrm{P}^4$ of degree $d$ with an isolated singularity are studied. If the singularity is of type $A_{2k+1}$, under the condition $d<(k+5)/2$, a classification of such hypersurfaces…
We construct algebraic surfaces with a large number of type A singularities. Bivariate polynomials presented in previous works for the construction of nodal surfaces and certain families of Belyi polynomials are used. In some cases explicit…
We prove that the determinantal complexity of a hypersurface of degree $d > 2$ is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least $5$. As a result, we obtain that the…
In this paper, we consider the following question: how many degree $d$ curves are there in $\mathbb{P}^3$ (passing through the right number of generic lines and points), whose image lies inside a $\mathbb{P}^2$, having $\delta$ nodes and…
In this paper we compute upper bounds for the number of ordinary triple points on a hypersurface in $P^3$ and give a complete classification for degree six (degree four or less is trivial, and five is elementary). But the real purpose is to…
We study parameter spaces of linear series on projective curves in the presence of unibranch singularities, i.e. {\it cusps}; and to do so, we stratify cusps according to value semigroup. We show that {\it generalized Severi varieties} of…
We show that the singularities of spacelike maximal surfaces in Lorentz-Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de…
We study the geometry of the smooth projective surfaces that are defined by Frobenius forms, a class of homogenous polynomials in prime characteristic recently shown to have minimal possible F-pure threshold among forms of the same degree.…
A monoid hypersurface is an irreducible hypersurface of degree d which has a singular point of multiplicity d-1. Any monoid hypersurface admits a rational parameterization, hence is of potential interest in computer aided geometric design.…
Let $X$ be an arbitrary smooth hypersurface in $\mathbb{C} \mathbb{P}^n$ of degree $d$. We prove the de Jong-Debarre Conjecture for $n \geq 2d-4$: the space of lines in $X$ has dimension $2n-d-3$. We also prove an analogous result for…