Michele Torielli
We investigate deformations of free and linear free divisors. We introduce a complex similar to the de Rham complex whose cohomology calculates deformation spaces. This cohomology turns out to be zero for many linear free divisors and to be constructible in many cases
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Motivated by a conjecture of Xiao, we study supporting divisors of fibred surfaces. On the one hand, after developing a formalism to treat one-dimensional families of varieties of any dimension, we give a structure theorem for fibred surfaces supported on relatively rigid divisors. On the other hand, we study how to produce supporting divisors by constructing a global adjoint map for a fibration over a curve (generalizing the infinitesimal constructions of Collino, Pirola, Rizzi and Zucconi).
In this note we characterize polarized parallel transport operators on irreducible holomorphic symplectic varieties which are deformations of generalized Kummer varieties. We then apply such characterization to show the existence of ample uniruled divisors on these varieties and derive some interesting consequences on their Chow group of 0-cycles.
Given a smooth, projective variety $X$ and an effective divisor $D\,\subseteq\, X$, it is well-known that the (topological) obstruction to the deformation of the fundamental class of $D$ as a Hodge class, lies in $H^2(\mathcal{O}_X)$. In this article, we replace $H^2(\mathcal{O}_X)$ by $H^2_D(\mathcal{O}_X)$ and give an analogous topological obstruction theory. We compare the resulting local topological obstruction theory with the geometric obstruction theory (i.e., the obstruction to the deformation of $D$ as an effective Cartier divisor of a first order infinitesimal deformations of $X$). We…
Under some conditions on the deformation type, which we expect to be satisfied for arbitrary irreducible symplectic varieties, we describe which big and nef line bundles on irreducible symplectic varieties have base divisors. In particular, we show that such base divisors are always irreducible and reduced. This is applied to understand the behaviour of divisorial base components of big and nef line bundles under deformations and for K3$^{[n]}$-type and Kum$^n$-type.
We study the deformation theory of rational curves on primitive symplectic varieties and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus in the universal locally trivial deformation. As applications, we extend Markman's deformation invariance of prime exceptional divisors along their Hodge locus to this singular framework and provide existence results for uniruled ample divisors on primitive symplectic varieties which are locally trivial deformations of any moduli space of semistable objects on a projective $K3$ or fibers of th…
Buchstaber and Mikhailov introduced the polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals on the basis of commuting vector fields on the symmetric square of hyperelliptic curves. In our previous paper, we constructed the field of meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3 and solutions of the systems for $g=3$ by these functions. In this paper, as an application of our previous results, we construct two parametric deformation of the KdV-hierarchy. This new system is integrated in the meromorphic functions on the sigma divisor of hyp…
We study linear free divisors, that is, free divisors arising as discriminants in prehomogeneous vector spaces, and in particular in quiver representation spaces. We give a characterization of the prehomogeneous vector spaces containing such linear free divisors. For reductive linear free divisors, we prove that the numbers of geometric and representation theoretic irreducible components coincide. As a consequence, we find that a quiver can only give rise to a linear free divisor if it has no (oriented or unoriented) cycles. We also deduce that the linear free divisors which appear in Sato and…
We introduce a variant of the usual Kahler forms on free and almost free divisors and their deformations, and show that they enjoy the same depth properties as Kahler forms on isolated complete intersection singularities. Using these forms, it is possible to describe analytically the vanishing cohomology in families of free divisors, in precise analogy with the classical description for the Milnor fibration of an ICIS, due to Brieskorn and Greuel. This applies in par- ticular to the family of discriminants of a versal deformation of an unstable map-germ.
We study deformations of pairs (X,D), with X smooth projective variety and D a smooth or a normal crossing divisor, defined over an algebraically closed field of characteristic 0. Using the differential graded Lie algebras theory and the Cartan homotopy construction, we are able to prove in a completely algebraic way the unobstructedness of the deformations of the pair (X,D) in many cases, e.g., whenever (X,D) is a log Calabi-Yau pair, in the case of a smooth divisor D in a Calabi Yau variety X and when D is a smooth divisor in |-m K_X|, for some positive integer m.
We study invariant divisors on the total spaces of the homogeneous deformations of rational complexity-one T-varieties constructed by Ilten and Vollmert. In particular, we identify a natural subgroup of the Picard group for any general fiber of such a deformation, which is canonically isomorphic to the Picard group of the special fiber. This isomorphism preserves Euler characteristic, intersection numbers, and the canonical class.
Linear free divisors are free divisors, in the sense of K.Saito, with linear presentation matrix (example: normal crossing divisors). Using techniques of deformation theory on representations of quivers, we exhibit families of linear free divisors as discriminants in representation spaces for real Schur roots of a finite quiver. We review some basic material on quiver representations, and explain in detail how to verify whether the discriminant is a free divisor and how to determine its components and their equations, using techniques of A. Schofield. As an illustration, the linear free diviso…
We introduce a method for obtaining new classes of free divisors from representations $V$ of connected linear algebraic groups $G$ where $\dim(G)=\dim(V)$, with $V$ having an open orbit. We give sufficient conditions that the complement of this open orbit, the "exceptional orbit variety", is a free divisor (or a slightly weaker free* divisor) for "block representations" of both solvable groups and extensions of reductive groups by them. These are representations for which the matrix defined from a basis of associated "representation vector fields" on $V$ has block triangular form, with blocks …
We prove that under certain conditions, the quantum cohomology of a positively monotone compact symplectic manifold is a deformation of the symplectic cohomology of the complement of a simple crossings symplectic divisor. We also prove rigidity results for the skeleton of the divisor complement.
Hyperbolic Embeddings in Toric Geometry: Effectivity and Deformation Stability
本文研究了托里几何中双曲嵌入的变形行为，通过有效方法分析例外集并证明其变形稳定性。
We introduce the concept of a prehomogeneous determinant as a nonreduced version of a linear free divisor. Both are special cases of prehomogeneous vector spaces. We show that the roots of the $b$-function are symmetric about -1 for reductive prehomogeneous determinants and for regular special linear free divisors. For general prehomogeneous determinants, we describe conditions under which this symmetry still holds. Combined with Kashiwara's theorem on the roots of b-functions, our symmetry result shows that -1 is the only integer root of the b-function. This gives a positive answer to a probl…
Let M be a compact hyperkaehler manifold, and W the coarse moduli of complex deformations of M. Every positive integer class v in $H^2(M)$ defines a divisor $D_v$ in W consisting of all algebraic manifolds polarized by v. We prove that every connected component of this divisor is dense in W.
We present versal complex analytic families, over a smooth base and of fibre dimension zero, one, or two, where the discriminant constitutes a free divisor. These families include finite flat maps, versal deformations of reduced curve singularities, and versal deformations of Gorenstein surface singularities in C^5. It is shown that such free divisors often admit a "fast normalization", obtained by a single application of the Grauert-Remmert normalization algorithm. For a particular Gorenstein surface singularity in C^5, namely the simple elliptic singularity of type \tilde A_4, we exhibit an …
Multiplicity-free products of Schubert divisors
研究任意域上旗簇中舒伯特除子类无重数乘积在 Chow 环中的次数，能将其分解为舒伯特簇类线性组合，通过研究组合系数给出判别准则，确定系数情况。
We show that there is an unexpected relation between free divisors and stability and coincidence thresholds for projective hypersurfaces.
Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings. We study the variety of unitary representations of the fundamental group of U with certain restrictions related to the divisor. We show that the possible singularities of this variety as well as of the corresponding moduli space of irreducible representations are quadratic. In the course of our proof we exhibit a differential graded Lie algebra of which reflects our deformation problem.
We describe two situations where adding the adjoint divisor to a divisor D with smooth normalization yields a free divisor. Both also involve stability or versality. In the first, D is the image of a corank one stable germ of a map from complex n-space to complex (n+1)-space, and is not free. In the second, D is the discriminant of a versal deformation of a weighted homogeneous function with isolated critical point (subject to certain numerical conditions on the weights). Here D itself is already free. We also prove an elementary result, inspired by these first two, from which we obtain a plet…
In this paper we classify mildly singular Fano varieties with maximal Picard number whose effective divisors are numerically eventually free.
We prove that any divisor as in the title must be normal crossing.
Given a smooth complex projective variety X, a line bundle L of X an element v of H^1(O_X) and a section s in H^0(L) that deforms to first order in the direction v, we give a sufficient condition on v in terms of Koszul cohomology for this first order deformation to extend to an analytic deformation. We apply this result to improve known results on the paracanonical system of a variety of maximal Albanese dimension, due to Beauville in the case of surfaces and to Lazarsfeld-Popa in higher dimension. In particular, we prove the inequality p_g(X)>=χ(K_X)+q(X)-1 for a variety X of maximal Albanes…
We investigate differential systems occurring in the study of particular non-isolated singularities, the so-called linear free divisors. We obtain a duality theorem for these D-modules taking into account filtrations, and deduce degeneration properties of certain Frobenius manifolds associated to linear sections of the Milnor fibres of the divisor.
We find explicit free resolutions for the $\scr D$-modules ${\scr D} f^s$ and ${\scr D}[s] f^s/{\scr D}[s] f^{s+1}$, where $f$ is a reduced equation of a locally quasi-homogeneous free divisor. These results are based on the fact that every locally quasi-homogeneous free divisor is Koszul free, which is also proved in this paper
We start by constructing a new root system for rational triple singularities and determine the number of roots for each rational triple singularity. Then we show that, for each root, we obtain a linear free divisor. So we obtain a new family of linear free divisors. This gives the converse part of an existing theorem which says, by using the quiver representation, that linear free divisors come from a tree. We prove that our construction is independent of the orientation on the rational triple trees. Furthermore, we deduce that linear free divisors defined by rational triple quivers satisfy th…
We discuss Bernstein polynomials of reductive linear free divisors. We define suitable Brieskorn lattices for these non-isolated singularities, and show the analogue of Malgrange's result relating the roots of the Bernstein polynomial to the residue eigenvalues on the saturation of these Brieskorn lattices.
A $K3$ surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that a similar statement holds for the generic couple $(X,H)$ with $X$ a deformation of $(K3)^{[n]}$ and $H$ an ample divisor of square 2 for Beauville's quadratic form. If $n=2$ then according to the conjecture $X$ is a double cover of a (singular) sextic 4-fold in $\PP^5$. It follows from the conjecture that a deformation of $(K3)^{[n]}$ carrying a divisor (not necessarily ample) of degree 2 has an anti-symplectic birational involution. We test the conjecture…
We reformulate base point free theorems. Our formulation is flexible and has some important applications. One of the main purposes of this paper is to prove a generalization of the base point free theorem in Fukuda's paper: On numerically effective log canonical divisors.
An infinite family of irreducible homogeneous free divisors in $K[x, y, z]$ is constructed. Indeed, we identify sets of monomials $X$ such that the general polynomial supported on $X$ is a free divisor.
On a metric graph we introduce the notion of a free divisor as a replacement for the notion of a base point free complete linear system on a curve. By means of an example we show that the Clifford inequality is the only obstruction for the existence of very special free divisors on a graph. This is different from the situation of base point free linear systems on curves. It gives rise to the existence of many types of divisors on graphs that cannot be lifted to curves maintaining the rank and it also shows that classifications made for linear systems of some fixed small positive Clifford index…
In this paper we prove that the Bernstein-Sato polynomial of any free divisor for which the $D[s]$-module $D[s] h^s$ admits a Spencer logarithmic resolution satisfies the symmetry property $b(-s-2) = \pm b(s)$. This applies in particular to locally quasi-homogeneous free divisors (for instance, to free hyperplane arrangements), or more generally, to free divisors of linear Jacobian type. We also prove that the Bernstein-Sato polynomial of an integrable logarithmic connection $E$ and of its dual $E^*$ with respect to a free divisor of linear Jacobian type are related by the equality $b_{E}(s)=\…
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 complex hypersurface D in complex affine n-space C^n is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for n at most 4. Analogous to Grothendieck's comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic differential forms computes the complex cohomology of the complement of D in C^n. We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the Lie algebra of linear logarithmic vector fields is r…
We introduce tautological system defined by prehomogenous 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.
We prove a structure theorem for the differential operator in the 0-term of the ${\cal V}$-filtration with respect to a free divisor. Using this theorem, we give a formula for the logarithmic de Rham complex in terms of ${\cal V}_0$-modules. We also give a sufficient condition for perversity of the logarithmic de Rham complex.
We study linear functions on fibrations whose central fibre is a linear free divisor. We analyse the Gauss-Manin system associated to these functions, and prove the existence of a primitive and homogenous form. As a consequence, we show that the base space of the semi-universal unfolding of such a function carries a Frobenius manifold structure.
In this paper we survey the role of D-module theory in the comparison between logarithmic and meromorphic de Rham complexes of integrable logarithmic connections with respect to free divisors, and we present some new linearity conditions on the Jacobian ideal which arise in this setting.
The relationship between ${\cal D}$-modules and free divisors has been studied in a general setting by L. Narváez and F.J. Calderón. Using the ideas of these works we prove in this article a duality formula between two ${\cal D}$-modules associated to a class of free divisors on ${\bf C}^n$ and we give some applications
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 \cite{hyparr}.
This paper has been withdrawn by the author since the proof of Lemma 8 is not correct.
A plane curve on a the projective space over a field of characteristic zero is free if its associated sheaf T of tangent vector fields tangent is a free module. Relatively few free curves are known. Here we prove that a divisor consisting of a union of curves of a pencil of plane projective curves with the same degree and with a smooth base locus is a free divisor if and only this union contains all the singular members of the pencil and its Jacobian ideal is locally a complete intersection.
In this note we present a description of wave front evolving from an algebraic hypersurface by means of a pull-back of the discriminantal loci of a tame polynomial via a polynomial mapping. As an application we give examples of wave fronts which define free/almost free divisors near the focal point.
Let $X$ be a nonsingular complex projective variety and $D$ a locally quasi-homogeneous free divisor in $X$. In this paper we study a numerical relation between the Chern class of the sheaf of logarithmic derivations on $X$ with respect to $D$, and the Chern-Schwartz-MacPherson class of the complement of $D$ in $X$. Our result confirms a conjectural formula for these classes, at least after push-forward to projective space; it proves the full form of the conjecture for locally quasi-homogeneous free divisors in $\mathbb P^n$. The result generalizes several previously known results. For example…
Advances on a conjecture about free divisors
本文通过研究强欧拉同次性和弱科兹尔自由性，证明了关于自由除子的猜想在新条件下成立，并推翻了线性自由除子的猜想。
In this paper infinite families of examples of irreducible free and nearly free curves in the complex projective plane which are not rational curves and whose local singularites can have an arbitrary number of branches are given. All these examples answer negatively to some conjectures proposed by A. Dimca and G. Sticlaru. Our examples say nothing about the most remarkable conjecture by A. Dimca and G. Sticlaru, i.e. every rational cuspidal plane curve is either free or nearly free.
Rational pairs, recently introduced by Kollár and Kovács, generalize rational singularities to pairs $(X,D)$. Here $X$ is a normal variety and $D$ is a reduced divisor on $X$. Integral to the definition of a rational pair is the notion of a thrifty resolution, also defined by Kollár and Kovács, and in order to work with rational pairs it is often necessary to know whether a given resolution is thrifty. In this paper we present several foundational results that are helpful for identifying thrifty resolutions and analyzing their behavior. We also show that general hyperplane sections of rational…
Let $φ:X\to S$ 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 $φ$ is a Cohen-Macaulay $\mathcal{O}_X$-module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $φ$, then $f\circ φ=0$ 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 $φ:\mathbb{C}^{n+1}\to \mathbb{C}^n$ with critical set of codimension $2$ has a $T^1_…
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.
The Chern class of the sheaf of logarithmic derivations along a simple normal crossing divisor equals the Chern-Schwartz-MacPherson class of the complement of the divisor. We extend this equality to more general divisors, which are locally analytically isomorphic to free hyperplane arrangements.
A 'prehomogeneous vector space' is a rational representation $ρ:G\to\mathrm{GL}(V)$ of a connected complex linear algebraic group $G$ that has a Zariski open orbit $Ω\subset V$. Mikio Sato showed that the hypersurface components of $D:=V\setminus Ω$ are related to the rational characters $H\to\mathrm{GL}(\mathbb{C})$ of $H$, an algebraic abelian quotient of $G$. Mimicking this work, we investigate the 'additive functions' of $H$, the homomorphisms $Φ:H\to (\mathbb{C},+)$. Each such $Φ$ is related to an 'additive relative invariant', a rational function $h$ on $V$ such that $h\circ ρ(g)-h=Φ(g)$…
We define the principal divisor of a free noncommuatative function. We use these divisors to compare the determinantal singularity sets of free noncommutative functions. We show that the divisor of a noncommutative rational function is the difference of two polynomial divisors. We formulate a nontrivial theory of cohomology, fundamental groups and covering spaces for tracial free functions. We show that the natural fundamental group arising from analytic continuation for tracial free functions is a direct sum of copies of $\mathbb{Q}$. Our results contrast the classical case, where the analogo…
A characterization of freeness for plane curves in terms of the Hilbert function of the associated Milnor algebra is given as well as many new examples of rational cuspidal curves which are free. Some stronger properties are stated as conjectures.
Let X be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of X is a lattice with respect to the Beauville-Bogomolov pairing. A divisor E on X is called a prime exceptional divisor, if E is reduced and irreducible and of negative Beauville-Bogomolov degree. Let E be a prime exceptional divisor on X. We first observe that associated to E is a monodromy involution of the integral cohomology of X, which acts on the second cohomology lattice as the reflection by the cohomology class of E (Theorem 1.1). We then specialize to the case that X is deformation equi…
Let $k$ be a perfect field of positive characteristic and $Z$ an effective Cartier divisor in the projective line over $k$ with complement $U$. In this note, we establish some results about the formal deformation theory of overconvergent isocrystals on $U$ with fixed "local monodromy" along $Z$. En route, we show that a Hochschild cochain complex governs deformations of a module over an arbitrary associative algebra. We also relate this Hochschild cochain complex to a de Rham complex in order to understand the deformation theory of a differential module over a differential ring.
We define a class of plane curves which are close to the free divisors and such that conjecturally it contains the class of rational cuspidal curves. Using a recent result by U. Walther we show that any unicuspidal rational curve with a unique Puiseux pair is either free or belongs to this class.
Cancellation-free version of the quantum $K$-theoretic divisor axiom for the flag manifold in the quasi-minuscule case
本文在准次小情况下证明了量子K-理论除数公理的无消去版本，通过移除Lenart-Naito-Sagaki-Xu公理中的消去项，为旗 manifold 的研究提供了新的工具。
Numerical characterization of the hard Lefschetz classes of dimension two, II: supercritical collections of free divisor classes
本文刻画了光滑射影簇上超临界自由除子类集合的 Lefschetz 型算子核空间，解决了 Shenfeld-van Handel 的公开问题，并给出了 Alexandrov-Fenchel 与 Khovanskii-Teissier 不等式极值的代数几何证明。
In this paper we use the results from the first part to compute the vanishing topology for matrix singularities based on certain spaces of matrices. We place the variety of singular matrices in a geometric configuration of free divisors which are the "exceptional orbit varieties" for repesentations of solvable groups. Because there are towers of representations for towers of solvable groups, the free divisors actually form a tower of free divisors $E_n$, and we give an inductive procedure for computing the vanishing topology of the matrix singularities. The inductive procedure we use is an ext…
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