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Alexandru Dimca

Publications and source records attributed to Alexandru Dimca.

At least 55 records · Page 3Linked to original sources
arXiv2021

Ramblings on the freeness of affine hypersurfaces

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 $\mathbb{P}^n$, then we relate first the Jacobian syzygies of $D$ and those of $X$. Then we introduce two types of freeness for an affine hypersurface $X$, and prove various relations between them and the freeness of the projective hypersurface $D$. We write down a proof of the folklore result saying that an affine hypersurface is free if and only if all of its singularities are free, in the sense of K. Saito's definition in the local setting. In particular, smooth affine hypersurfaces and affine plane curves are always free. Some other results, involving global Tjurina numbers and minimal degrees of non trivial syzygies are also explored.

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arXiv2021

On the Bounded Negativity Conjecture and singular plane curves

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 conjectures as a replacement. We also develop bounds on numerical characteristics of curves constraining their negativity. For example, we show that the $H$-constant of a rational curve $C$ with at most $9$ singular points satisfies $H(C)>-2$ regardless of the characteristic.

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arXiv2020

Waring rank of symmetric tensors, and singularities of some projective hypersurfaces

We show that if a homogeneous polynomial $f$ in $n$ variables has Waring rank $n+1$, then the corresponding projective hypersurface $f=0$ has at most isolated singularities, and the type of these singularities is completely determined by the combinatorics of a hyperplane arrangement naturally associated with the Waring decomposition of $f$. We also discuss the relation between the Waring rank and the type of singularities on a plane curve, when this curve is defined by the suspension of a binary form, or when the Waring rank is 5.

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arXiv2020

Waring rank of binary forms, harmonic cross-ratio and golden ratio

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 used to show that the combinatorics of a line arrangement in the complex projective plane does not determine the Waring rank of the defining equation even in very simple situations.

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arXiv2020

On the Hilbert vector of the Jacobian module of a plane curve

We identify several classes of curves $C:f=0$, for which the Hilbert vector of the Jacobian module $N(f)$ can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational irreducible components. A result due to Hartshorne, on the cohomology of some rank 2 vector bundles on $\mathbb{P}^2$, is used to get a sharp lower bound for the initial degree of the Jacobian module $N(f)$, under a semistability condition.

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arXiv2020

Freeness and invariants of rational plane curves

Given a parameterization $ϕ$ of a rational plane curve C, we study some invariants of C via $ϕ$. We first focus on the characterization of rational cuspidal curves, in particular we establish a relation between the discriminant of the pull-back of a line via $ϕ$, the dual curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via $ϕ$, we prove that the (nearly) freeness of a rational curve can be tested by inspecting the Hilbert function of the kernel of a canonical map. As a by product, we also show that the global Tjurina number of a rational curve can be computed directly from one of its parameterization, without relying on the computation of an equation of C.

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arXiv2020

Unexpected curves in $\mathbb{P}^2$, line arrangements, and minimal degree of Jacobian relations

We reformulate a fundamental result due to Cook, Harbourne, Migliore and Nagel on the existence and irreduciblity of unexpected plane curves of a set of points $Z$ in $\mathbb{P}^2$, using the minimal degree of a Jacobian syzygy of the defining equation for the dual line arrangement $\mathcal A_Z$. Several applications of this new approach are given. In particular, we show that the irreducible unexpected quintics may occur only when the set $Z$ has the cardinality equal to 11 or 12, and describe five cases where this happens.

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arXiv2019

On complex supersolvable line arrangements

We show that the number of lines in an $m$--homogeneous supersolvable line arrangement is upper bounded by $3m-3$ and we classify the $m$--homogeneous supersolvable line arrangements with two modular points up-to lattice-isotopy. A lower bound for the number of double points $n_2$ in an $m$--homogeneous supersolvable line arrangement of $d$ lines is also considered. When $3 \leq m \leq 5$, or when $m \geq \frac{d}{2}$, or when there are at least two modular points, we show that $n_2 \geq \frac{d}{2}$, as conjectured by B. Anzis and S. O. Toh\u aneanu. This conjecture is shown to hold also for supersolvable line arrangements obtained as cones over generic line arrangements, or cones over arbitrary line arrangements having a generic vertex.

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arXiv2019

Saturation of Jacobian ideals: some applications to nearly free curves, line arrangements and rational cuspidal plane curves

In this note we describe the minimal resolution of the ideal $I_f$, the saturation of the Jacobian ideal of a nearly free plane curve $C:f=0$. In particular, it follows that this ideal $I_f$ can be generated by at most 4 polynomials. Related general results by Hassanzadeh and Simis on the saturation of codimension 2 ideals are discussed in detail. Some applications to rational cuspidal plane curves and to line arrangements are also given.

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arXiv2019

Higher order Jacobians, Hessians and Milnor algebras

We introduce and study higher order Jacobian ideals, higher order and mixed Hessians, higher order polar maps, and higher order Milnor algebras associated to a reduced projective hypersurface. We relate these higher order objects to some standard graded Artinian Gorenstein algebras, and we study the corresponding Hilbert functions and Lefschetz properties.

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arXiv2019

Addition-deletion results for the minimal degree of logarithmic derivations of arrangements

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 arrangements in a lot of new situations. Then, starting with Ziegler's example of a pair of arrangements of $d=9$ lines with $n_3=6$ triple points in addition to some double points, having the same combinatorics, but distinct minimal degree of a logarithmic derivation, we construct new examples of such pairs, for any number $d\geq 9$ of lines, and any number $n_3\geq 6$ of triple points. Moreover, we show that such examples are not possible for line arrangements having only double and triple points, with $n_3 \leq 5$.

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arXiv2019

Plane curves with three syzygies, minimal Tjurina curves curves, and nearly cuspidal curves

We start the study of reduced complex projective plane curves, whose Jacobian syzygy module has 3 generators. Among these curves one finds the nearly free curves introduced by the authors, and the plus-one generated line arrangements introduced by Takuro Abe. All the Thom-Sebastiani type plane curves, and more generally, any curve whose global Tjurina number is equal to a lower bound given by A. du Plessis and C.T.C. Wall, are 3-syzygy curves. Rational plane curves which are nearly cuspidal, i.e. which have only cusps except one singularity with two branches, are also related to this class of curves.

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arXiv2019

Jacobian syzygies and plane curves with maximal global Tjurina numbers

First we give a sharp upper bound for the cardinal $m$ of a minimal set of generators for the module of Jacobian syzygies of a complex projective reduced plane curve $C$. Next we discuss the sharpness of an upper bound, given by A. du Plessis and C.T.C. Wall, for the global Tjurina number of such a curve $C$, in terms of its degree $d$ and of the minimal degree $r\leq d-1$ of a Jacobian syzygy. We give a homological characterization of the curves whose global Tjurina number equals the du Plessis-Wall upper bound, which implies in particular that for such curves the upper bound for $m$ is also attained. Finally we prove the existence of curves with maximal global Tjurina numbers for certain pairs $(d,r)$. Moreover, we conjecture that such curves exist for any pair $(d,r)$, and that, in addition, they may be chosen to be line arrangements when $r\leq d-2$. This conjecture is proved for degrees $d \leq 11$.

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arXiv2019

The 0-th Fitting ideal of the Jacobian module of a plane curve

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 curves, whose global Tjurina number equals an upper bound given by A. du Plessis and C.T.C. Wall.

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arXiv2019

Versality, bounds of global Tjurina numbers and logarithmic vector fields along hypersurfaces with isolated singularities

We recall first the relations between the syzygies of the Jacobian ideal of the defining equation for a projective hypersurface $V$ with isolated singularities and the versality properties of $V$, as studied by du Plessis and Wall. Then we show how the bounds on the global Tjurina number of $V$ obtained by du Plessis and Wall lead to substantial improvements of our previous results on the stability of the reflexive sheaf $T\langle V\rangle$ of logarithmic vector fields along $V$, and on the Torelli property in the sense of Dolgachev-Kapranov of $V$.

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arXiv2019

Line and rational curve arrangements, and Walther's inequality

There are two invariants associated to any line arrangement: the freeness defect $ν(C)$ and an upper bound for it, denoted by $ν'(C)$, coming from a recent result by Uli Walther. We show that $ν'(C)$ is combinatorially determined, at least when the number of lines in $C$ is odd, while the same property is conjectural for $ν(C)$. In addition, we conjecture that the equality $ν(C)=ν'(C)$ holds if and only if the essential arrangement $C$ of $d$ lines has either a point of multiplicity $d-1$, or has only double and triple points. We prove both conjectures in some cases, in particular when the number of lines is at most 10. We also extend a result by H. Schenck on the Castenuovo-Mumford regularity of line arrangements to arrangements of possibly singular rational curves.

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arXiv2018

On the jumping lines of bundles of logarithmic vector fields along plane curves

For a reduced curve $C:f=0$ in the complex projective plane $\mathbb{P}^2$, we study the set of jumping lines for the rank two vector bundle $T\langle C \rangle $ on $\mathbb{P}^2$, whose sections are the logarithmic vector fields along $C$. We point out the relations of these jumping lines with the Lefschetz type properties of the Jacobian module of $f$ and with the Bourbaki ideal of the module of Jacobian syzygies of $f$. In particular, when the vector bundle $T\langle C \rangle $ is unstable, a line is a jumping line if and only if it meets the 0-dimensional subscheme defined by this Bourbaki ideal, a result going back to Schwarzenberger. Other classical general results by Barth, Hartshorne and Hulek resurface in the study of this special class of rank two vector bundles.

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