Towards transversality of singular varieties: splayed divisors

We study a natural generalization of transversally intersecting smooth hypersurfaces in a complex manifold: hypersurfaces, whose components intersect in a transversal way but may be themselves singular. Such hypersurfaces will be called splayed divisors. A splayed divisor is characterized by a property of its Jacobian ideal. This yields an effective test for splayedness. Two further characterizations of a splayed divisor are shown, one reflecting the geometry of the intersection of its components, the other one using K. Saito's logarithmic derivations. As an application we prove that a union of smooth hypersurfaces has normal crossings if and only if it is a free divisor and has a radical Jacobian ideal. Further it is shown that the Hilbert-Samuel polynomials of splayed divisors satisfy a certain additivity property.Comment: 15 pages, 1 figure; v2: minor revision: inaccuracies corrected and references updated; v3: final version, to appear in Publ. RIM

Splayed divisors and their Chern classes

We obtain several new characterizations of splayedness for divisors: a Leibniz property for ideals of singularity subschemes, the vanishing of a `splayedness' module, and the requirements that certain natural morphisms of modules and sheaves of logarithmic derivations and logarithmic differentials be isomorphisms. We also consider the effect of splayedness on the Chern classes of sheaves of differential forms with logarithmic poles along splayed divisors, as well as on the Chern-Schwartz-MacPherson classes of the complements of these divisors. A postulated relation between these different notions of Chern class leads to a conjectural identity for Chern-Schwartz-MacPherson classes of splayed divisors and subvarieties, which we are able to verify in several template situations.Comment: 18 pages, 1 figure. v2: minor inaccuracies corrected, references adde

Swallowtail on the shore

Platonic solids, Felix Klein, H.S.M. Coxeter and a flap of a swallowtail: The five Platonic solids tetrahedron, cube, octahedron, icosahedron and dodecahedron have always attracted much curiosity from mathematicians, not only for their sheer beauty but also because of their many symmetry properties. In this snapshot we will start from these symmetries, move on to groups, singularities, and finally find the connection between a tetrahedron and a “swallowtail”. Our running example is the tetrahedron, but every construction can be carried out with any other of the Platonic solids

Normal crossings in local analytic geometry

Das Hauptziel dieser Dissertation ist eine effektive algebraische Charakterisierung von Divisoren (= Hyperflächen) mit normalen Kreuzungen in komplexen Mannigfaltigkeiten anzugeben. Um eine derartige Charakterisierung zu finden, studieren wir sowohl logarithmische Vektorfelder entlang eines Divisors, d.h., Vektorfelder des umgebenden Raumes, die in allen glatten Punkten des Divisors tangential an ihn sind, als auch logarithmische Differentialformen. Mit Hilfe der zugehörigen Theorie, entwickelt von K. Saito, wird eine Charakterisierung von Divisoren mit normalen Kreuzungen durch logarithmische Differentialformen (Vektorfelder) gezeigt. Des weiteren wird eine Charakterisierung durch das logarithmische Residuum vorgestellt (diese beruht auf Ergebnissen von Granger und Schulze). Damit kann eine Frage von K. Saito beantwortet werden. Im zweiten Kapitel werden Singularitäten eines Divisors mit normalen Kreuzungen untersucht, insbesondere betrachten wir das Jacobi Ideal, das den singulären Ort des Divisors definiert. Unser Hauptsatz besagt, dass ein Divisor genau dann normale Kreuzungen in einem Punkt besitzt, wenn er frei in diesem Punkt, sein Jacobi Ideal radikal und seine Normalisierung Gorenstein ist. Freie Divisoren werden durch logarithmische Differentialformen definiert und bilden eine Klasse von Divisoren, die insbesondere Divisoren mit normalen Kreuzungen enthält. Da eine algebraische Charakterisierung von freien Divisoren durch deren Jacobi Ideale existiert (nach A. G. Aleksandrov), ergibt sich aus unserem Resultat eine rein algebraische Charakterisierung der normalen Kreuzungsbedingung. Im Laufe des Beweises des Hauptsatzes werden gespreizte Divisoren eingeführt, die eine leichte Verallgemeinerung von Divisoren mit normalen Kreuzungen darstellen. Im letzten Teil der Arbeit werden weiterreichende Probleme betrachtet: Zuerst fragen wir, welche radikalen Ideale Jacobi Ideale von Divisoren sein können. Dann werden gespreizte Divisoren genauer untersucht, insbesondere zeigen wir, dass ihre Hilbert-Samuel Polynome eine gewisse Additivitätsbedingung erfüllen. Schließlich wird eine weitere Verallgemeinerung von Divisoren mit normalen Kreuzungen betrachtet, sogenannte Mikado Divisoren. Hier charakterisieren wir ebene Mikado Kurven durch ihr Jacobi Ideal.The main objective of this thesis is to give an effective algebraic characterization of normal crossing divisors (= hypersurfaces) in complex manifolds. In order to obtain such a characterization we study logarithmic vector fields along a divisor, i.e., vector fields defined on the ambient space, which are tangent to the divisor at its smooth points, as well as logarithmic differential forms. Using the corresponding theory, which was developed by K. Saito, a characterization of a normal crossing divisor in terms of logarithmic differential forms (vector fields) is shown. Also a characterization of a normal crossing divisor in terms of the logarithmic residue is given (which is essentially due to Granger and Schulze). With this a question posed by K. Saito in 1980 can be answered. In the second chapter we study singularities of normal crossing divisors, in particular we consider Jacobian ideals, which define the singular locus of a divisor. The main theorem is that a divisor has normal crossings at point if and only if it is free at the point, its Jacobian ideal is radical and its normalization is Gorenstein. Free divisors are defined via logarithmic vector fields and form a class of divisors containing normal crossing divisors. Since there exists an algebraic characterization of free divisors by their Jacobian ideals, our result yields a purely algebraic characterization of the normal crossings property. During the proof of the main theorem splayed divisors are introduced, which are a slight generalization of normal crossing divisors. In the last part we consider further-reaching questions: first we ask, which radical ideals can be Jacobian ideals of divisors. Then splayed divisors are studied in more detail, in particular, we show that their Hilbert-Samuel polynomials satisfy a certain additivity property. Finally, we consider another generalization of normal crossing divisors, so-called mikado divisors. Here the plane curve case is studied and we characterize mikado curves by their Jacobian ideal

Matrix Factorizations of the discriminant of SnS_n

Consider the symmetric group $S_n$ acting as a reflection group on the polynomial ring $k[x_1, \ldots, x_n]$, where $k$ is a field such that Char$(k)$ does not divide $n!$. We use Higher Specht polynomials to construct matrix factorizations of the discriminant of this group action: these matrix factorizations are indexed by partitions of $n$ and respect the decomposition of the coinvariant algebra into isotypical components. The maximal Cohen-Macaulay modules associated to these matrix factorizations give rise to a noncommutative resolution of the discriminant and they correspond to the nontrivial irreducible representations of $S_n$. All our constructions are implemented in Macaulay2 and we provide several examples. We also discuss extensions of these results to Young subgroups of $S_n$.Comment: 23 pages, comments welcome

Classification of singularities of cluster algebras of finite type

We provide a complete classification of the singularities of cluster algebras of finite type. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields of arbitrary characteristic. Furthermore, from the same perspective, we study a family of cluster algebras, which are not of finite type and which arise from a star shaped quiver.Comment: 36 page

Grassmannian categories of infinite rank

We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded maximal Cohen-Macaulay modules over a certain hypersurface singularity. We show that generically free modules of rank $1$ in a Grassmannian category of infinite rank are in bijection with the Pl\"ucker coordinates in an appropriate Grassmannian cluster algebra of infinite rank. In particular, we show that this bijection is structure preserving, as it relates rigidity in the category to compatibility of Pl\"ucker coordinates. Along the way, we develop a combinatorial formula to compute the dimension of the $\mathrm{Ext}^1$-spaces between any two generically free modules of rank $1$ in the Grassmannian category of infinite rank