arXiv subjects
Publications and source records attributed to Gabriel Sticlaru.
We show that all the possible pairs of integers occur as exponents for free or nearly free irreducible plane curves and line arrangements, by producing only two types of simple families of examples. The topology of the complements of these curves and line arrangements is also discussed, and many of them are shown not to be $K(π,1)$ spaces.
We investigate the relationship among several numerical invariants associated to a (free) projective hypersurface $V$: the sequence of mixed multiplicities of its Jacobian ideal, the Hilbert polynomial of its Milnor algebra, and the sequence of exponents when $V$ is free. As a byproduct, we obtain explicit equations for some of the homaloidal surfaces in the projective 3-dimensional space constructed by C. Ciliberto, F. Russo and A. Simis.
We define a class of plane curves which are close to the free divisors and such that conjecturally it contains the class of rational cuspidal curves. Using a recent result by U. Walther we show that any unicuspidal rational curve with a unique Puiseux pair is either free or belongs to this class.
A characterization of freeness for plane curves in terms of the Hilbert function of the associated Milnor algebra is given as well as many new examples of rational cuspidal curves which are free. Some stronger properties are stated as conjectures.
Using the minors in Hessian matrices, we introduce new graded algebras associated to a homogeneous polynomial. When the associated projective hypersurface has isolated singularities, these algebras are related to some new local algebras associated to isolated hypersurface singularities, which generalize their Tjurina algebras. One consequence of our results is a new way to determine the number of weighted homogeneous singularities of such a hypersurface.
Let $V$ be a projective hypersurface having only isolated singularities. We show that these singularities are weighted homogeneous if and only if the Koszul syzygies among the partial derivatives of an equation for $V$ are exactly the syzygies with a generic first component vanishing on the singular locus subscheme of $V$. This yields in particular a positive answer in this setting to a question raised by Morihiko Saito and the first author. Finally we explain how our result can be used to improve the listing of Jacobian syzygies of a given degree by a computer algebra system such as Singular, CoCoA or Macaulay2.
We investigate the 0-th local cohomology of the Jacobian ring of a homogeneous polynomial defining a projective hypersurface whose singular locus is a 0-dimensional complete intersection.
We investigate the relation between codimension two smooth complete intersections in a projective space and some naturally associated graded algebras. We give some examples of log-concave polynomials and we propose two conjectures for these algebras.
We show that there is an unexpected relation between free divisors and stability and coincidence thresholds for projective hypersurfaces.
In this paper we present some properties for projective hypersurfaces, smooth and singular, to be criteria for identification. To make the decision with these criteria, we have included procedures written in Singular language.
This paper continues our researches \cite{DS1, DS2, DS3} by computing some invariants based on Hilbert-Poincaré series associated to Milnor algebras. Our computations are for some of the classical surfaces and 3-folds with different configurations of isolated singularities. As a by-product of a recent result of E. Sernesi, we give examples of classical hypersurfaces which are (or are not) projectively rigid. We also include a Singular program to compute the invariants and to decide if a singular projective hypersurface is nodal and projectively rigid.
We study the interplay between the cohomology of the Koszul complex of the partial derivatives of a homogeneous polynomial $f$ and the pole order filtration $P$ on the cohomology of the open set $U=\PP^n \setminus D$, with $D$ the hypersurface defined by $f=0$. The relation is expressed by some spectral sequences, which may be used on one hand to determine the filtration $P$ in many cases for curves and surfaces, and on the other hand to obtain information about the syzygies involving the partial derivatives of the polynomial $f$. The case of a nodal hypersurface $D$ is treated in terms of the defects of linear systems of hypersurfaces of various degrees passing through the nodes of $D$. When $D$ is a nodal surface in $\PP^3$, we show that $F^2H^3(U) \ne P^2H^3(U)$ as soon as the degree of $D$ is at least 4.
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 expressed in terms of defects or superabundances. The case of Chebyshev hypersurfaces is considered as a test for this result and leads to a potentially infinite family of nodal hypersurfaces having nontrivial Alexander polynomials.
First we construct a free resolution for the Milnor (or Jacobian) algebra $M(f)$ of a complex projective Chebyshev plane curve $\CC_d:f=0$ of degree $d$. In particular, this resolution implies that the dimensions of the graded components $M(f)_k$ are constant for $k \geq 2d-3.$ Then we show that the Milnor algebra of a nodal plane curve $C$ has such a behaviour if and only if all the irreducible components of $C$ are rational. For the Chebyshev curves, all of these components are in addition smooth, hence they are lines or conics and explicit factorizations are given in this case.