arXiv subjects

Gabriel Sticlaru

Publications and source records attributed to Gabriel Sticlaru.

At least 19 recordsLinked to original sources

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

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

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 $τ(C)$ 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 $τ(C)$ 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

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

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

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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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

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

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

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

On the Alexander polynomials of conic-line arrangements

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 pencils of index $k\geq 2$. Then we apply them to the Hesse arrangement of conics and to some of its degenerations. The results are completed by computations using computer algebra. In particular, we construct conic-line arrangements which are non-reduced pencil-type arrangements and have as roots of their Alexander polynomials roots of unity of order 7. Such roots are not known and are conjectured not to exist in the class of line arrangements.

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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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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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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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