arXiv subjects

Eleonore Faber

Publications and source records attributed to Eleonore Faber.

29 records · Page 2Linked to original sources

arXiv2019

Non-Commutative Resolutions of Toric Varieties

Let $R$ be the coordinate ring of an affine toric variety. We show that the endomorphism ring $End_R(\mathbb A),$ where $\mathbb A$ is the (finite) direct sum of all (isomorphism classes of) conic $R$-modules, has finite global dimension. Furthermore, we show that $End_R(\mathbb A)$ is a non-commutative crepant resolution if and only if the toric variety is simplicial. For toric varieties over a perfect field $k$ of prime characteristic, we show that the ring of differential operators $D_\mathsf{k}(R)$ has finite global dimension.

math.AC↗

arXiv2018

Conway-Coxeter friezes and mutation: a survey

In this survey article we explain the intricate links between Conway-Coxeter friezes and cluster combinatorics. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the frieze changes under cluster mutation. Moreover, we provide a combinatorial formula for the number of submodules of a string module, and with that a simple way to compute the frieze associated to a fixed cluster tilting object in a cluster category of Dynkin type $A$ in the sense of Caldero and Chapoton.

math.RA↗

arXiv2018

Noncommutative resolutions of discriminants

We give an introduction to the McKay correspondence and its connection to quotients of $\mathbb{C}^n$ by finite reflection groups. This yields a natural construction of noncommutative resolutions of the discriminants of these reflection groups. This paper is an extended version of E.F.'s talk with the same title delivered at the ICRA.

math.AG↗

arXiv2017

Mutation of friezes

We study mutations of Conway-Coxeter friezes which are compatible with mutations of cluster-tilting objects in the associated cluster category of Dynkin type $A$. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the frieze changes under cluster mutation. We observe how the frieze can be divided into four distinct regions, relative to the entry at which we want to mutate, where any two entries in the same region obey the same mutation rule. Moreover, we provide a combinatorial formula for the number of submodules of a string module, and with that a simple way to compute the frieze associated to a fixed cluster-tilting object in a cluster category of Dynkin type $A$ in the sense of Caldero and Chapoton.

math.RA↗

arXiv2015

Computing global dimension of endomorphism rings via ladders

This paper deals with computing the global dimension of endomorphism rings of maximal Cohen--Macaulay (=MCM) modules over commutative rings. Several examples are computed. In particular, we determine the global spectra, that is, the sets of all possible finite global dimensions of endomorphism rings of MCM-modules, of the curve singularities of type $A_n$ for all $n$, $D_n$ for $n \leq 13$ and $E_{6,7,8}$ and compute the global dimensions of Leuschke's normalization chains for all ADE curves, as announced in [Dao-Faber-Ingalls]. Moreover, we determine the centre of an endomorphism ring of a MCM-module over any curve singularity of finite MCM-type. In general, we describe a method for the computation of the global dimension of an endomorphism ring $\mathrm{End}_R M$, where $R$ is a Henselian local ring, using $\mathrm{add}(M)$-approximations. When $M\neq 0$ is a MCM-module over $R$ and $R$ is Henselian local of Krull dimension $\leq 2$ with a canonical module and of finite MCM-type, we use Auslander--Reiten theory and Iyama's ladder method to explicitly construct these approximations.

math.RT↗

arXiv2014

Characterizing normal crossing hypersurfaces

The objective of this article is to give an effective algebraic characterization of normal crossing hypersurfaces in complex manifolds. It is shown that a hypersurface has normal crossings if and only if it is a free divisor, has a radical Jacobian ideal and a smooth normalization. Using K. Saito's theory of free divisors, also a characterization in terms of logarithmic differential forms and vector fields is found and and finally another one in terms of the logarithmic residue using recent results of M. Granger and M. Schulze.

math.AG↗

arXiv2014

Noncommutative (crepant) desingularizations and the global spectrum of commutative rings

In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but subsumes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introduce and study the global spectrum of a ring $R$, that is, the set of all possible finite global dimensions of endomorphism rings of MCM $R$-modules. Finally, we use a variety of methods to compute global dimension for many endomorphism rings.

math.AC↗

arXiv2014

Chern classes of splayed intersections

We generalize the Chern class relation for the transversal intersection of two nonsingular varieties to a relation for possibly singular varieties, under a 'splayedness' assumption. The relation is shown to hold for both the Chern-Schwartz-MacPherson class and the Chern-Fulton class. The main tool is a formula for Segre classes of splayed subschemes. We also discuss the Chern class relation under the assumption that one of the varieties is a general very ample divisor.

math.AG↗

arXiv2013

Measuring Singularities with Frobenius: The Basics

Consider a polynomial $f$ defined over a field $k$, the multiplicity is perhaps the most naive measurement of the singularities of $f$. This paper describes the first steps toward understanding a much more subtle measure of singularities which arises naturally in three different contexts-- analytic, algebro-geometric, and finally, algebraic. Miraculously, all three approaches lead to essentially the same measurement of singularities: the log canonical threshold (in characteristic zero) and the closely related $F$-pure threshold (in characteristic $p$). In this paper we present only the first steps in understanding these invariants, with an emphasis on the prime characteristic setting.

math.AG↗

arXiv2013

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.

math.AG↗

arXiv2012

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.

math.AG↗