arXiv subjects

Alexandru Dimca

Publications and source records attributed to Alexandru Dimca.

At least 37 records · Page 2Linked to original sources

arXiv2025

Some remarks on plane curves related to freeness

Let $C$ be a reduced complex projective plane curve, and let $d_1$ and $d_2$ be the first two smallest exponents of $C$. For a free curve $C$ of degree $d$, there is a simple formula relating $d,d_1, d_2$ and the total Tjurina number of $C$. Our first result discusses how this result changes when the curve $C$ is no longer free. For a free line arrangement, the Poincaré polynomial coincides with the Betti polynomial $B(t)$ and with the product $P(t)=(1+d_1t)(1+d_2t)$. Our second result shows that for any curve $C$, the difference $P(t)-B(t)$ is a polynomial $a t +bt^2$, with $a$ and $b$ non-negative integers. Moreover $a =0$ or $b=0$ if and only if $C$ is a free line arrangement. Finally we give new bounds for the second exponent $d_2$ of a line arrangement $\mathcal A$, the corresponding lower bound being an improvement of a result by H. Schenck concerning the relation between the maximal exponent of $\mathcal A$ and the maximal multiplicity of points in $\mathcal A$.

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arXiv2024

Construction of free curves by adding osculating conics to a given cubic curve

In the present article we construct new families of free and nearly free curves starting from a plane cubic curve $C$ and adding some of its hyperosculating conics. We present results that involve nodal cubic curves and the Fermat cubic. In addition, we provide new insight into the geometry of the $27$ hyperosculating conics of the Fermat cubic curve using well-chosen group actions.

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arXiv2024

Plus-one generated curves, Briançon-type polynomials and eigenscheme ideals

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 fibers of some Briançon-type polynomials constructed by E. Artal Bartolo, Pi. Cassou-Noguès and I. Luengo Velasco. They yield new examples of free, nearly free or minimal plus-one generated curves, as well as counter-examples to the conjecture saying that a supersolvable curve is free. In the final section we give a characterization of plus-one generated curves in terms of eigenscheme ideals, similar to the characterization of free curves given by R. Di Gennaro, G. Ilardi, R.M. Miró-Roig, H. Schenck and J. Vallès in a recent paper. Then we apply this result to the construction of minimal plus-one generated curves obtained by putting together at least two members in a pencil of curves related to Briançon-type polynomials.

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arXiv2024

Curves with Jacobian syzygies of the same degree

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, this class of curves seems to be surprisingly large. In particular, any line arrangement $\mathcal A$ of $d=2k+1\geq 5 $ lines having only double and triple points is in this class if the number of triple points is $k$ and if they are all situated on a line $L \in \mathcal A$, see Theorem 6.2

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arXiv2024

Koszul complexes and spectra of projective hypersurfaces with isolated singularities

For a projective hypersurface $Z$ with isolated singularities, we generalize some well-known assertions in the nonsingular case due to Griffiths, Scherk, Steenbrink, Varchenko, and others about the relations between the Steenbrink spectrum, the Poincaré polynomial of the Jacobian ring, and the roots of Bernstein-Sato polynomial for a defining polynomial $f$ up to sign forgetting the multiplicities. We have to use the pole order spectrum and the alternating sum of the Poincaré series of certain subquotients of the Koszul cohomologies, and study the pole order spectral sequence. We show sufficient conditions for vanishing or non-vanishing of the differential $d_1$ of the spectral sequence, which are useful in many applications. We prove also symmetries of the dimensions of the subquotients of Koszul cohomologies, which are crucial for computing the roots of BS polynomials. We can deduce that the roots of BS polynomial whose absolute values are larger than $n-1-n/d$ are determined by the ``torsion part" of the Jacobian ring (modulo the roots of BS polynomial for $Z$) if all the singularities of $Z$ are weighted homogeneous. Here $d=°f$ and $n$ is the dimension of the ambient affine space.

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arXiv2024

On the birationality of the Hessian maps of quartic curves and cubic surfaces

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 proof of the similar property for cubic surfaces, which is already known by the work of these two authors.

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arXiv2024

On the Castelnuovo-Mumford regularity of curve arrangements

The Castelnuovo-Mumford regularity of the Jacobian algebra and of the graded module of derivations associated to a general curve arrangement in the complex projective plane are studied. The key result is an addition-deletion type result, similar to results obtained by H. Schenck, H. Terao, S. Tohaneanu and M. Yoshinaga, but in which no quasi homogeneity assumption is needed.

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arXiv2024

Hunting for Miyaoka-Kobayashi curves

We study the possible singularities of Miyaoka-Kobayashi curves and prove several results about the non-existence of such curves in degrees $\geq 8$.

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arXiv2024

From Pascal's Theorem to the geometry of Ziegler's line arrangements

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. Independently, Sergey Yuzvinsky has arrived in 1993 at the same type of line arrangements in order to show that formality is not determined by the combinatorics. In this note we look into the geometry of such line arrangements, and find out an unexpected relation to the classical Pascal's Theorem. Our results give information on the minimal degree of a Jacobian syzygy and on the formality of such hexagonal line arrangements in general, without an explicit choice for the six vertices of the hexagon.

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arXiv2023

On free curves and related open problems

In this paper we collect the main properties of free curves in the complex projective plane and a lot of conjectures and open problems, both old and new. In the quest to understand the mystery of free curves, many tools were developed and many results were obtained, which apply to any reduced plane curve, and some of them are recorded here.

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arXiv2023

On free and plus-one generated curves arising from free curves by addition-deletion of a line

In a recent paper, after introducing the notion of plus-one generated hyperplane arrangements, Takuro Abe has shown that if we add (resp. delete) a line to (resp. from) a free line arrangement, then the resulting line arrangement is either free or plus-one generated. In this note we prove that the same properties hold when we replace the line arrangement by a free curve and add (resp. delete) a line. The proof uses a new version of a key result due originally to H. Schenck, H. Terao and M. Yoshinaga, in which no quasi homogeneity assumption is needed. Two conjectures about the Tjurina number of a union of two plane curve singularities are also stated. As a geometric application, we show that, under a mild numerical condition, the projective closure of a contractible, irreducible affine plane curve is either free or plus-one generated, using a deep result due to U. Walther.

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arXiv2023

Lefschetz properties of Jacobian algebras and Jacobian modules

Let $V:f=0$ be a hypersurface of degree $d \geq 3$ in the complex projective space $\mathbb{P}^n$, $n \geq 3$, having only isolated singularities. Let $M(f)$ be the associated Jacobian algebra and $H: \ell=0$ be a hyperplane in $\mathbb{P}^n$ avoiding the singularities of $V$, but such that $V \cap H$ is singular. We related the Lefschetz type properties of the linear maps $\ell: M(f)_k \to M(f)_{k+1}$ induced by the multiplication by linear form $\ell$ to the singularities of the hyperplane section $V \cap H$. Similar results are obtained for the Jacobian module $N(f)$.

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arXiv2023

On the duals of smooth projective complex hypersurfaces

We show first that a generic hypersurface $V$ of degree $d\geq 3$ in the complex projective space $ \mathbb{P}^n$ of dimension $n \geq 3$ has at least one hyperplane section $V \cap H$ containing exactly $n$ ordinary double points, alias $A_1$ singularities, in general position, and no other singularities. Equivalently, the dual hypersurface $V^{\vee}$ has at least one normal crossing singularity of multiplicity $n$. Using this result, we show that the dual of any smooth hypersurface with $n,d \geq 3$ has at least a very singular point $q$, in particular a point $q$ of multiplicity $\geq n$.

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arXiv2023

Construction of free curves by adding lines to a given curve

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 singularities of $C$. These lines $L$ have in common that the intersection $C \cap L$ consists of a small number of points. We introduce the notion of a supersolvable plane curve and conjecture that such curves are always free, as in the known case of line arrangements. Some evidence for this conjecture is given as well, both in terms of a general result in the case of quasi homogeneous singularities and in terms of specific examples. We construct a new example of maximizing curve in degree 8 and the first and unique known example of maximizing curve in degree 9. In the final section, we use a stronger version of a result due to Schenck, Terao and Yoshinaga to construct families of free conic-line arrangements by adding lines to the conic-line arrangements of maximal Tjurina number recently classified by V. Beorchia and R. M. Miró-Roig in arXiv:2303.04665.

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arXiv2023

Maximizing curves viewed as free curves

The aim of this paper is to provide a direct link between maximizing curves that occur in the construction of smooth algebraic surfaces having the maximal possible Picard numbers and reduced free plane curves with simple singularities. We also investigate odd degree plane curves with simple singularities having maximal total Tjurina number.

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arXiv2022

Waring ranks of sextic binary forms via geometric invariant theory

We determine the Waring ranks of all sextic binary forms with complex coefficients using a Geometric Invariant Theory approach. Using the five basic invariants for sextic binary forms, our results give a rapid method to determine the Waring rank of any given sextic binary form. In particular, we shed new light on a claim by E. B. Elliott at the end of the 19th century concerning the binary sextics with Waring rank 3. We show that for binary forms of arbitrary degree the cactus rank, a.k.a. scheme rank, is determined by the corresponding Waring rank. Finally we determine the border ranks of all binary sextics.

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arXiv2022

On plane conic arrangements with nodes and tacnodes

In the present paper, we study arrangements of smooth plane conics having only nodes and tacnodes as the singularities. We provide an interesting estimation on the number of nodes and tacnodes that depends only on a linear function of the number of conics. Based on that result, we obtain a new upper bound on the number of tacnodes which turns out to be better than Miyaoka's bound for a large enough number of conics. We also study the freeness and nearly freeness of such arrangements providing a detailed description.

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arXiv2021

The Hessian polynomial and the Jacobian ideal of a reduced hypersurface in $\mathbb{P}^n$

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 the regularity of $M(f)$ when $V(f)$ has a positive dimensional singular locus. In certain situations, we prove that the regularity is bounded by $(d-2)(n+1)$, which is the degree of the Hessian polynomial of $f$. However, this is not always the case, and we prove that in $\mathbb{P}^n$ the regularity of the Milnor algebra can grow quadratically in $d$.

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