Gabriel Sticlaru
YOU?
Author Swipe
View article: Bourbaki modules and the module of Jacobian derivations of projective hypersurfaces
Bourbaki modules and the module of Jacobian derivations of projective hypersurfaces Open
Two properties of projective hypersurfaces related to the module of Jacobian derivations, namely being tame and being plus-one generated, are discussed in this paper. Tame hypersurfaces are related to Bourbaki ideals, and free hypersurface…
View article: Plus-one generated curves, Briançon-type polynomials and eigenscheme ideals
Plus-one generated curves, Briançon-type polynomials and eigenscheme ideals Open
We define the minimal plus-one generated curves and prove a result explaining why they are the closest relatives of the free curves, after the nearly free curves. Then we look at the projective closures of the general and of the special fi…
View article: Curves with Jacobian syzygies of the same degree
Curves with Jacobian syzygies of the same degree Open
In this notes we study complex projective plane curves whose graded module of Jacobian syzygies is generated by its minimal degree component. Examples of such curves include the smooth curves as well as the maximal Tjurina curves. However,…
View article: From Pascal's Theorem to the geometry of Ziegler's line arrangements
From Pascal's Theorem to the geometry of Ziegler's line arrangements Open
Günter Ziegler has shown in 1989 that some homological invariants associated with the free resolutions of Jacobian ideals of line arrangements are not determined by combinatorics. His classical example involves hexagons inscribed in conics…
View article: Construction of Free Curves by Adding Lines to a Given Curve
Construction of Free Curves by Adding Lines to a Given Curve Open
In the present note we construct new families of free plane curves starting from a curve C and adding high order inflectional tangent lines of C , lines joining the singularities of the curve C , or lines in the tangent cone of some singul…
View article: On the Alexander polynomials of conic-line arrangements
On the Alexander polynomials of conic-line arrangements Open
In the present paper we compute Alexander polynomials for certain classes of conic-line arrangements in the complex projective plane which are related to pencils. We prove two general results for curve arrangements coming from Halphen penc…
View article: On the birationality of the Hessian maps of quartic curves and cubic surfaces
On the birationality of the Hessian maps of quartic curves and cubic surfaces Open
We show that the Hessian map of quartic plane curves is a birational morphism onto its image, thus bringing new evidence for a very interesting conjecture of Ciro Ciliberto and Giorgio Ottaviani. Our new approach also yields a simpler proo…
View article: Construction of free curves by adding lines to a given curve
Construction of free curves by adding lines to a given curve Open
In the present note we construct new families of free plane curves starting from a curve $C$ and adding high order inflectional tangent lines of $C$, lines joining the singularities of the curve $C$, or lines in the tangent cone of some si…
View article: Waring Ranks of Sextic Binary Forms via Geometric Invariant Theory
Waring Ranks of Sextic Binary Forms via Geometric Invariant Theory Open
View article: Waring rank of binary forms, harmonic cross-ratio and golden ratio
Waring rank of binary forms, harmonic cross-ratio and golden ratio Open
We discuss the Waring rank of binary forms of degree 4 and 5, without\nmultiple factors, and point out unexpected relations to the harmonic\ncross-ratio, j-invariants and the golden ratio. These computations of ranks for\nbinary forms are …
View article: On the Bounded Negativity Conjecture and Singular Plane Curves
On the Bounded Negativity Conjecture and Singular Plane Curves Open
There are no known failures of Bounded Negativity in characteristic 0. In the light of recent work showing the Bounded Negativity Conjecture fails in positive characteristics for rational surfaces, we propose new characteristic free conjec…
View article: Addition-deletion results for the minimal degree of a Jacobian syzygy of a union of two curves
Addition-deletion results for the minimal degree of a Jacobian syzygy of a union of two curves Open
Let $C:f=0$ be a reduced curve in the complex projective plane. The minimal degree $mdr(f)$ of a Jacobian syzygy for $f$, which is the same as the minimal degree of a derivation killing $f$, is an important invariant of the curve $C$, for …
View article: On the birationality of the Hessian maps of quartic curves and cubic surfaces
On the birationality of the Hessian maps of quartic curves and cubic surfaces Open
We show that the hessian map of quartic plane curves is a birational morphism onto its image, thus bringing new evidence for a very interesting conjecture of Ciro Ciliberto and Giorgio Ottaviani. Our new approach also yields a simpler proo…
View article: The Hessian polynomial and the Jacobian ideal of a reduced hypersurface in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">P</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math>
The Hessian polynomial and the Jacobian ideal of a reduced hypersurface in Open
View article: Ramblings on the freeness of affine hypersurfaces
Ramblings on the freeness of affine hypersurfaces Open
In this note we look at the freeness for complex affine hypersurfaces. If $X \subset \mathbb{C}^n$ is such a hypersurface, and $D$ denotes the associated projective hypersurface, obtained by taking the closure of $X$ in $\mathbb{P}^n$, the…
View article: Ramblings on the freeness of affine plane curves
Ramblings on the freeness of affine plane curves Open
In this note we look at the freeness for complex affine plane curves. If $X$
is such a curve, and $C$ denotes the associated projective plane curve,
obtained by taking the closure of $X$, then we relate the Jacobian syzygies of
$C$ and tho…
View article: On the Bounded Negativity Conjecture and singular plane curves
On the Bounded Negativity Conjecture and singular plane curves Open
There are no known failures of Bounded Negativity in characteristic 0. In the light of recent work showing the Bounded Negativity Conjecture fails in positive characteristics for rational surfaces, we propose new characteristic free conjec…
View article: Addition–deletion results for the minimal degree of logarithmic derivations of hyperplane arrangements and maximal Tjurina line arrangements
Addition–deletion results for the minimal degree of logarithmic derivations of hyperplane arrangements and maximal Tjurina line arrangements Open
View article: Waring Rank of Symmetric Tensors, and Singularities of Some Projective Hypersurfaces
Waring Rank of Symmetric Tensors, and Singularities of Some Projective Hypersurfaces Open
View article: On the jumping lines of bundles of logarithmic vector fields along plane curves
On the jumping lines of bundles of logarithmic vector fields along plane curves Open
For a reduced curve $C:f=0$ in the complex projective plane $\\mathbb{P}^2$,\nwe study the set of jumping lines for the rank two vector bundle $T\\langle C\n\\rangle $ on $\\mathbb{P}^2$, whose sections are the logarithmic vector fields\na…
View article: Waring rank of binary forms, harmonic cross-ratio and golden ratio
Waring rank of binary forms, harmonic cross-ratio and golden ratio Open
We discuss the Waring rank of binary forms of degree 4 and 5, without multiple factors, and point out unexpected relations to the harmonic cross-ratio, j-invariants and the golden ratio. These computations of ranks for binary forms are use…
View article: Computing Milnor fiber monodromy for some projective hypersurfaces
Computing Milnor fiber monodromy for some projective hypersurfaces Open
We describe an algorithm computing the monodromy and the pole order filtration on the top Milnor fiber cohomology of hypersurfaces in $\mathbb{P}^n$ whose pole order spectral sequence degenerates at the second page. In the case of hyperpla…
View article: Deformations of plane curves and Jacobian syzygies
Deformations of plane curves and Jacobian syzygies Open
We relate the equianalytic and the equisingular deformations of a reduced complex plane curve to the Jacobian syzygies of its defining equation. Several examples and conjectures involving rational cuspidal curves are discussed.
View article: Freeness and invariants of rational plane curves
Freeness and invariants of rational plane curves Open
International audience
View article: The Hessian polynomial and the Jacobian ideal of a reduced hypersurface in $\mathbb{P}^n$
The Hessian polynomial and the Jacobian ideal of a reduced hypersurface in $\mathbb{P}^n$ Open
For a reduced hypersurface $V(f) \subseteq \mathbb{P}^n$ of degree $d$, the Castelnuovo-Mumford regularity of the Milnor algebra $M(f)$ is well understood when $V(f)$ is smooth, as well as when $V(f)$ has isolated singularities. We study t…
View article: The Hessian polynomial and the Jacobian ideal of a reduced homogeneous polynomial
The Hessian polynomial and the Jacobian ideal of a reduced homogeneous polynomial Open
The end of the Jacobian module and the Castelnuovo-Mumford regularity of the
Milnor algebra associated to a smooth hypersurface or to a singular
hypersurface with isolated singularities are well understood. In this note we
discuss these tw…
View article: Line and rational curve arrangements, and Walther’s inequality
Line and rational curve arrangements, and Walther’s inequality Open
There are two invariants associated to any line arrangement: the freeness defect \nu(C) and an upper bound for it, denoted by \nu'(C) , coming from a recent result by Uli Walther. We show that \nu'(C) is combinatorially determined, at leas…
View article: Addition-deletion results for the minimal degree of logarithmic derivations of arrangements
Addition-deletion results for the minimal degree of logarithmic derivations of arrangements Open
We study the change of the minimal degree of a logarithmic derivation of a hyperplane arrangement under the addition or the deletion of a hyperplane, and give a number of applications. First, we prove the existence of Tjurina maximal line …
View article: The 0-th Fitting ideal of the Jacobian module of a plane curve
The 0-th Fitting ideal of the Jacobian module of a plane curve Open
We describe the 0-th Fitting ideal of the Jacobian module of a plane curve in terms of determinants involving the Jacobian syzygies of this curve. This leads to new characterizations of maximal Tjurina curves, that is of non free plane cur…
View article: Saturation of Jacobian ideals: Some applications to nearly free curves, line arrangements and rational cuspidal plane curves
Saturation of Jacobian ideals: Some applications to nearly free curves, line arrangements and rational cuspidal plane curves Open