Free Divisors Research Articles

Algebraic curves and foliations

Consider a field k ${\mathsf {k}}$ of characteristic 0 $\hskip.001pt 0$ , not necessarily algebraically closed, and a fixed algebraic curve f = 0 $f=0$ defined by a tame polynomial f ∈ k [ x , y ] $f\in {\mathsf {k}}[x,y]$ with only quasi-homogeneous singularities. We prove that the space of holomorphic foliations in the plane A k 2 $\mathbb {A}^2_{\mathsf {k}}$ having f = 0 $f=0$ as a fixed invariant curve is generated as k [ x , y ] ${\mathsf {k}}[x,y]$ -module by at most four elements, three of them are the trivial foliations f d x , f d y $fdx,fdy$ and d f $df$ . Our proof is algorithmic and constructs the fourth foliation explicitly. Using Serre's GAGA and Quillen–Suslin theorem, we show that for a suitable field extension K ${\mathsf {K}}$ of k ${\mathsf {k}}$ such a module over K [ x , y ] ${\mathsf {K}}[x,y]$ is actually generated by two elements, and therefore, such curves are free divisors in the sense of K. Saito. After performing Groebner basis for this module, we observe that in many well-known examples, K = k ${\mathsf {K}}={\mathsf {k}}$ .

On an Equivalence of Divisors on M¯0,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\overline {\ ext {M}}_{0,n}$\\end{document} from Gromov-Witten Theory and Conformal Blocks

We consider a conjecture that identifies two types of base point free divisors on M¯0,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\overline {\ ext {M}}_{0,n}$\\end{document}. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated with simple Lie algebras in type A. Here we reduce this conjecture on M¯0,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\overline {\ ext {M}}_{0,n}$\\end{document} to the same statement for n = 4. A reinterpretation leads to a proof of the conjecture on M¯0,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\overline {\ ext {M}}_{0,n}$\\end{document} for a large class, and we give sufficient conditions for the non-vanishing of these divisors.

Fano manifolds with many free divisors

Fano manifolds with many free divisors

Conic-Line Arrangements in the Complex Projective Plane

The main goal of this note is to begin a systematic study on conic-line arrangements in the complex projective plane. We show a de Bruijn–Erdős type inequality and Hirzebruch-type inequality for a certain class of conic-line arrangements having ordinary singularities. We will also study, in detail, certain conic-line arrangements in the context of the geography of log-surfaces and free divisors in the sense of Saito.

Hodge ideals of free divisors

We consider the Hodge filtration on the sheaf of meromorphic functions along free divisors for which the logarithmic comparison theorem holds. We describe the Hodge filtration steps as submodules of the order filtration on a cyclic presentation in terms of a special factor of the Bernstein–Sato polynomial of the divisor and we conjecture a bound for the generating level of the Hodge filtration. Finally, we develop an algorithm to compute Hodge ideals of such divisors and we apply it to some examples.

Bernstein–Sato varieties and annihilation of powers

Given a complex germ $f$ near the point $\mathfrak{x}$ of the complex manifold $X$, equipped with a factorization $f = f_{1} \cdots f_{r}$, we consider the $\mathscr{D}_{X,\mathfrak{x}}[s_{1}, \dots, s_{r}]$-module generated by $ F^{S} := f_{1}^{s_{1}} \cdots f_{r}^{s_{r}}$. We show for a large class of germs that the annihilator of $F^{S}$ is generated by derivations and this property does not depend on the chosen factorization of $f$. We further study the relationship between the Bernstein-Sato variety attached to $F$ and the cohomology support loci of $f$, via the $\mathscr{D}_{X,\mathfrak{x}}$-map $\nabla_{A}$. This is related to multiplication by $f$ on certain quotient modules. We show that for our class of divisors the injectivity of $\nabla_{A}$ implies its surjectivity. Restricting to reduced, free divisors, we also show the reverse, using the theory of Lie-Rinehart algebras. In particular, we analyze the dual of $\nabla_{A}$ using techniques pioneered by Narvaez-Macarro. As an application of our results we establish a conjecture of Budur in the tame case: if $\text{V}(f)$ is a central, essential, indecomposable, and tame hyperplane arrangement, then the Bernstein-Sato variety associated to $F$ contains a certain hyperplane. By the work of Budur, this verifies the Topological Mulivariable Strong Monodromy Conjecture for tame arrangements. Finally, in the reduced and free case, we characterize local systems outside the cohomology support loci of $f$ near $\mathfrak{x}$ in terms of the simplicity of modules derived from $F^{S}.$

Lazarsfeld-Mukai Bundles of Rank 2 on a Polarized K3 Surface of Low Genus

Let X be a K3 surface and let H be a very ample line bundle on X of sectional genus g ≤ 9. In this paper, we characterize the destabilizing sheaf of the Lazarsfeld-Mukai bundle EC,Z of rank 2 associated with a smooth curve C ∈ |H| and a base point free divisor Z on C with h0(OC(Z)) = 2, in the case where it is not H-slope stable.

Characterizations of freeness for equidimensional subspaces

The purpose of this paper is to investigate properties of the minimal free resolution of the modules of multi-logarithmic forms along a reduced equidimensional subspace. We first consider a notion of freeness for reduced complete intersections, and more generally for reduced equidimensional subspaces embedded in a smooth manifold, which generalizes the notion of Saito free divisors. The first main result is a characterization of freeness in terms of the projective dimension of the module of multi-logarithmic k -forms, where k is the codimension. We also prove that there is a perfect pairing between the module of multi-logarithmic differential k -forms and the module of multi-logarithmic k -vector fields which generalizes the duality between the corresponding modules in the hypersurface case. We deduce from this perfect pairing a duality between the Jacobian ideal and the module of multi-residues of multi-logarithmic k -forms. In the last part of this paper, we investigate logarithmic modules along some examples of free singularities. The main result in this section is an explicit computation of the minimal free resolution of the module of multi-logarithmic forms and multi-residues for quasi-homogeneous complete intersection curves which uses our first main theorem.

Tautological systems and free divisors

We introduce tautological systems defined by prehomogeneous actions of reductive algebraic groups. If the complement of the open orbit is a linear free divisor satisfying a certain finiteness condition, we show that these systems underly mixed Hodge modules. A dimensional reduction is considered and gives rise to one-dimensional differential systems generalizing the quantum differential equation of projective spaces.

Chern classes of logarithmic derivations for free divisors with Jacobian ideal of linear type

Let $X$ be a nonsingular variety defined over an algebraically closed field of characteristic $0$, and $D$ be a free divisor with Jacobian ideal of linear type. We compute the Chern class of the sheaf of logarithmic derivations along $D$ and compare it with the Chern–Schwartz–MacPherson class of the hypersurface complement. Our result establishes a conjecture by Aluffi raised in [Alu12b].

Triple root systems, rational quivers and examples of linear free divisors

The dual resolution graphs of rational triple point (RTP) singularities can be seen as a generalization of Dynkin diagrams. In this work, we study the triple root systems corresponding to those diagrams. We determine the number of roots for each RTP singularity, and show that for each root we obtain a linear free divisor. Furthermore, we deduce that linear free divisors defined by rational triple quivers with roots in the corresponding triple root systems satisfy the global logarithmic comparison theorem. We also discuss a generalization of these results to the class of rational singularities with almost reduced Artin cycle.

A Module-theoretic Characterization of Algebraic Hypersurfaces

AbstractIn this note we prove the following surprising characterization: if X ⊂ is an (embedded, non-empty, proper) algebraic variety deûned over a field k of characteristic zero, then X is a hypersurface if and only if the module of logarithmic vector fields of X is a reflexive -module. As a consequence of this result, we derive that if is a free -module, which is shown to be equivalent to the freeness of the t-th exterior power of for some (in fact, any) t ≤ n, then necessarily X is a Saito free divisor.

Free and Nearly Free Curves vs. Rational Cuspidal Plane Curves

We define a class of plane curves that are close to the free divisors in terms of the local cohomology of their Jacobian algebras and such that, conjecturally, any rational cuspidal curve C is either free or belongs to this class. We prove this conjecture when the degree of C is either even or a prime power, or when the group of C is abelian.

K-theoretic Defect in Chern Class Identity for a Free Divisor

Let $X$ be a nonsingular variety defined over an algebraically closed field of characteristic $0$, and $D$ be a free divisor. We study the motivic Chern class of $D$ in the Grothendieck group of coherent sheaves $G_0(X)$, and another class defined by the sheaf of logarithmic differentials along $D$. We give explicit calculations of the difference of these two classes when: $D$ is a divisor on a nonsingular surface; $D$ is a hyperplane arrangement whose affine cone is free.

A residual duality over Gorenstein rings with application to logarithmic differential forms

Kyoji Saito's notion of a free divisor was generalized by the first author to reduced Gorenstein spaces and by Delphine Pol to reduced Cohen-Macaulay spaces. Starting point is the Aleksandrov-Terao theorem: A hypersurface is free if and only if its Jacobian ideal is maximal Cohen-Macaulay. Pol obtains a generalized Jacobian ideal as a cokernel by dualizing Aleksandrov's multi-logarithmic residue sequence. Notably it is essentially a suitably chosen complete intersection ideal that is used for dualizing. Pol shows that this generalized Jacobian ideal is maximal Cohen-Macaulay if and only if the module of Aleksandrov's multi-logarithmic differential k-forms has (minimal) projective dimension k-1, where k is the codimension in a smooth ambient space. This equivalent characterization reduces to Saito's definition of freeness in case k=1. In this article we translate Pol's duality result in terms of general commutative algebra. It yields a more conceptual proof of Pol's result and a generalization involving higher multi-logarithmic forms and generalized Jacobian modules.

Chern classes of logarithmic vector fields for locally quasi-homogeneous free divisors

Chern classes of logarithmic vector fields for locally quasi-homogeneous free divisors

Quasihomogeneous free divisors with only normal crossings in codimension one

We prove that any divisor as in the title must be normal crossing.

Homological properties of determinantal arrangements

We explore a natural extension of braid arrangements in the context of determinantal arrangements. We show that these determinantal arrangements are free divisors. Additionally, we prove that free determinantal arrangements defined by the minors of 2×n matrices satisfy nice combinatorial properties.We also study the topology of the complements of these determinantal arrangements, and prove that their higher homotopy groups are isomorphic to those of S3. Furthermore, we find that the complements of arrangements satisfying those same combinatorial properties above have Poincaré polynomials that factor nicely.

On the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus. II: The general case

We provide a general formula and give an explicit expression of the Lefschetz zeta function for any quasi-unipotent map on the n-dimensional torus. The Lefschetz zeta function is used to characterize the minimal set of Lefschetz periods for Morse–Smale diffeomorphisms on the n-dimensional torus; we completely describe this set, for different families containing infinitely many Morse–Smale diffeomorphisms. Moreover we show that for any given odd integer, there are Morse–Smale diffeomorphisms such that the corresponding minimal set of Lefschetz periods consists of all square free divisors of the given number. The results of the present article generalize the previous results of Berrizbeitia–Sirvent [7]. The methods used are based on the arithmetical properties of the number n.

Lifting free divisors

Let $\varphi:X\to S$ be a morphism between smooth complex analytic spaces, and let $f=0$ define a free divisor on $S$. We prove that if the deformation space $T^1_{X/S}$ of $\varphi$ is a Cohen-Macaulay $\mathcal{O}_X$-module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $\varphi$, then $f\circ \varphi=0$ defines a free divisor on $X$; this is generalized in several directions. Among applications we recover a result of Mond-van Straten, generalize a construction of Buchweitz-Conca, and show that a map $\varphi:\mathbb{C}^{n+1}\to \mathbb{C}^n$ with critical set of codimension $2$ has a $T^1_{X/S}$ with the desired properties. Finally, if $X$ is a representation of a reductive complex algebraic group $G$ and $\varphi$ is the algebraic quotient $X\to S=X// G$ with $X// G$ smooth, we describe sufficient conditions for $T^1_{X/S}$ to be Cohen-Macaulay of codimension $2$. In one such case, a free divisor on $\mathbb{C}^{n+1}$ lifts under the operation of "castling" to a free divisor on $\mathbb{C}^{n(n+1)}$, partially generalizing work of Granger-Mond-Schulze on linear free divisors. We give several other examples of such representations.