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Alexandru Dimca

Publications and source records attributed to Alexandru Dimca.

At least 19 recordsLinked to original sources
arXiv2026

A nine-line counterexample to a conjecture on the minimal degree of Jacobian relations

We construct two arrangements of nine lines in the complex projective plane with isomorphic intersection lattices but with different minimal degrees of Jacobian relations. The common weak combinatorics is \[ (n_2,n_3,n_4)=(9,7,1), \] so the example is not the classical Ziegler-Yuzvinsky pair, whose weak combinatorics is $(n_{2},n_{3}) = (18,6)$. For the two defining equations $f$ and $g$ we prove \[ {\rm mdr}(f)=4,\qquad {\rm mdr}(g)=5. \] Since the degree is $d=9$, the first equality gives ${\rm mdr}(f)<d/2$. Hence the pair gives a counterexample to the Generalized Terao Conjecture.

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arXiv2026

Plane curve singularities and Fitting ideals

In this note we investigate the Fitting ideals associated to the Tjurina ideal of a non quasi-homogeneous plane curve singularity. Special properties occur when the difference between Milnor number and Tjurina number is at most 2.

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arXiv2026

On Ziegler pairs of line arrangements: from non-existence to abundance

We study Ziegler pairs of line arrangements from both numerical and homological perspectives. We distinguish classical Ziegler pairs, for which the minimal degree of a Jacobian relation denoted by ${\rm mdr}$ differs, from the more general notion detected by distinct graded Betti data. First, we show that for arrangements of $d<9$ lines the intersection lattice determines ${\rm mdr}$, and hence classical Ziegler pairs do not occur in this range. Then we list several distinct Ziegler pairs with $d=10$ in the broader sense. In particular, we construct higher-degree examples with the same intersection lattice, the same minimal degree of a Jacobian relation, and the same Hilbert function of the Milnor algebra, but with different minimal graded free resolutions. Another rather surprising consequence is that being maximal Tjurina for a line arrangement is not determined by the combinatorics.

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arXiv2026

On the Jacobian algebras of Ziegler pairs of plane arrangements

We consider a Ziegler pair of plane arrangements, that is two plane arrangements $\mathcal{A}:f=0$ and $\mathcal{A}':f'=0$ in the projective space $\mathbb{P}^3$, such that the intersection lattices $L(\mathcal{A})$ and $L(\mathcal{A}')$ are isomorphic, but the Betti numbers of the minimal resolutions of their Jacobian algebras are not the same. We introduce several properties for such pairs and relate them to cones over Ziegler pairs of line arrangements in $\mathbb{P}^2$.

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arXiv2026

On the degree of the singular subscheme of hypersurfaces in ${\mathbb P}^n$

Explicit formulas determining the dimension and the degree of the singular subscheme of hypersurfaces in ${\mathbb P}^n$ are given in terms of the graded Betti numbers of the minimal free resolution of the corresponding Jacobian algebra. This gives in particular new restrictions which must be satisfied by such graded Betti numbers. We define a homologically strictly plus-one generated hypersurface, and show that such a hypersurface has a singular locus of dimension $n-2$ under some conditions.

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arXiv2026

Graded Betti numbers of the Jacobian algebra of surfaces in $\mathbb P^3$

We compute an explicit closed formula for the Hilbert polynomial of the Jacobian algebra $M(f)$ of a reduced surface $X:f=0$ in $\mathbb P^3$ in terms of the graded Betti numbers of the algebra $M(f)$. When $X$ has only isolated singularities, two results by A. du Plessis and C. T. C. Wall yield new necessary conditions for a set of positive integers to be the graded Betti numbers of the Jacobian algebra of such a surface. The comparison with the plane curve case is discussed in detail and additional information is given in the case of nodal surfaces. A natural conjecture on the smallest 4 exponents of $X$ is stated and support for it is provided. In the final section we construct four natural Jacobian syzygies for surfaces $X$ coming from pencils of surfaces.

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arXiv2026

Graded Betti numbers of the Jacobian algebra and total Tjurina numbers of plane curves

In this paper we compute an explicit closed formula for the total Tjurina number $\tau(C)$ of a reduced projective plane curve $C$ in terms of the graded Betti numbers of the corresponding Jacobian algebra. This formula allows a completely new view point on the classical upper bounds for the total Tjurina number $\tau(C)$ of a plane curve $C$ given by A. du Plessis and C. T. C. Wall. This approach yields in particular a new necessary condition for a set of positive integers to be the graded Betti numbers of the Jacobian algebra of a reduced plane curve.

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arXiv2026

On type three complex plane curves

The type of a complex projective plane curve has been recently introduced by T. Abe, P. Pokora and the first author. In the same paper they have studied the type two curves. In this paper we study plane curves of type three, with special attention to low degree curves and line arrangements.

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arXiv2025

A minimal resolution for the Jacobian ideal of a generic curve arrangement

We consider a nodal curve $C$ in the complex projective plane whose irreducible components $C_i$ are smooth. A minimal set of generators $G$ for the first and second syzygy modules of the Jacobian ideal of $C$ are described, using recent results by Th. Kahle, H. Schenck, B. Sturmfels and M. Wiesmann on the likelihood correspondence. The elements of $G$ have explicit formulas in terms of the equations $f_i=0$ of the irreducible components $C_i$ of $C$. Similar results, including extensions to hypersurfaces arrangements in $\mathbb{P}^n$ were obtained by R. Burity, Z. Ramos, A. Simis and St. Toh\u aneanu with a genericity assumption which may not be easy to test in practice.

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arXiv2025

On the Jacobian syzygies for generic toric models

To a generic hypersurface in the affine torus $(\mathbb{C}^*)^n$ we associate a hypersurface arrangement in the projective space $\mathbb{P}^n$ consisting of the $n+1$ coordinate hyperplanes and a generic hypersurface, and compute the minimal graded resolutions of the corresponding Jacobian algebra.

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arXiv2025

Bourbaki modules and the module of Jacobian derivations of projective hypersurfaces

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 hypersurfaces are the simplest examples of tame hypersurfaces. Plus-one generated hypersurfaces are the non free hypersurfaces which are closest to the free ones, and it is an open question whether all of them are tame.

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arXiv2025

Free line arrangements with low maximal multiplicity

Let $\A$ be a free arrangement of $d$ lines in the complex projective plane, with exponents $d_1\leq d_2$. Let $m$ be the maximal multiplicity of points in $\A$. In this note, we describe first the simple cases $d_1 \leq m$. Then we study the case $d_1=m+1$, and describe which line arrangements can occur by deleting or adding a line to $\A$. When $d \leq 14$, there are only two free arrangements with $d_1=m+2$, namely one with degree $13$ and the other with degree $14$. We study their geometries in order to deepen our understanding of the structure of free line arrangements in general.

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arXiv2025

On the module of derivations of a line arrangement

To each multiple point $p$ in a line arrangement $ \mathcal A$ in the complex projective plane we associate a local derivation $\tilde D_p \in D_0( \mathcal A)$. We show first that these derivations span the graded module of derivations $D_0( \mathcal A)$ in all degrees $\geq d -3$, where $d$ is the number of lines in $ \mathcal A$, see Theorem 1.4 and Theorem 1.6. Then, to each local derivation $\tilde D_p \in D_0( \mathcal A)$ we associate a polynomial $g_p$ which seems to play a key role in the characterization of the freeness of $ \mathcal A$, see Theorem 1.10, as well as in the study of the position of the multiple points of $ \mathcal A$ with respect to unions of lines, see Corollary 1.13 and Conjecture 1.14. Corollary 1.9 gives a result of an independent interest, namely a lower bound for the maximal exponent of a plane curve having a line as an irreducible component.

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arXiv2025

Quasi-homogeneous singularities of projective hypersurfaces and Jacobian syzygies

We prove an unexpected general relation between the Jacobian syzygies of a projective hypersurface $V\subset \mathbb{P}^n$ with only isolated singularities and the nature of its singularities. This allows to establish a new method for the identification of quasi-homogeneous hypersurface isolated singularities. The result gives an insight on how the geometry is reflected in the Jacobian syzygies and extends previous results of the first, second and last author for free and nearly free plane curves [1].

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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 Poincar\'e 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

A new hierarchy for complex plane curves

We define the type of a plane curve as the initial degree of the corresponding Bourbaki ideal. Then we show that this invariant behaves well with respect to the union of curves. Curves of type $0$ are precisely the free curves, while curves of type $1$ are the plus-one generated curves. In this paper, we first show that line arrangements and conic-line arrangements can exhibit all the theoretically possible types. In the second part, we study the properties of the curves of type $2$ and construct families of line arrangements and conic-line arrangements of this type.

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arXiv2024

Plus-one generated curves, Brian\c{c}on-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 Brian\c{c}on-type polynomials constructed by E. Artal Bartolo, Pi. Cassou-Nogu\`es 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. Mir\'o-Roig, H. Schenck and J. Vall\`es 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 Brian\c{c}on-type polynomials.

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