Eleonore Faber

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 generalizes 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.

TOWARDS TRANSVERSALITY OF SINGULAR VARIETIES: SPLAYED DIVISORS

We study a natural generalization of transversally intersecting smooth hypersurfaces in a complex manifold: hypersurfaces, whose compo- nents intersect in a transversal way but may be themselves singular. Such hypersurfaces will be called splayed divisors. A splayed divisor is charac- terized by a property of its Jacobian ideal. This yields an eective test for splayedness. Two further characterizations of a splayed divisor are shown, one reecting the geometry of the intersection of its components, the other one us- ing K. Saitos 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.

Today’s menu: Geometry and resolution of singular algebraic surfaces

The courses are Triviality, Tangency, Transversality, Symmetry, Simplicity, Singularity. These characteristic local plates serve as our invitation to algebraic surfaces and their resolution. Please take a seat.

Measuring Singularities with Frobenius: The Basics

The multiplicity is an important first step in measuring singularities, but it is too crude to give a good measurement. 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).

Noncommutative Resolutions of Discriminants

We give an introduction to the McKay correspondence and its connection to quotients of Cn 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.