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Mathias Schulze

Publications and source records attributed to Mathias Schulze.

At least 37 records · Page 2Linked to original sources
arXiv2018

Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals

Integration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a specific type of syzygies which correspond to logarithmic vector fields along the Gram determinant formed of the independent external and loop momenta. We present an explicit generating set of solutions in Baikov representation, valid for any number of loops and external momenta, obtained from the Laplace expansion of the Gram determinant. We provide a rigorous mathematical proof that this set of solutions is complete. This proof relates the logarithmic vector fields in question to ideals of submaximal minors of the Gram matrix and makes use of classical resolutions of such ideals.

hep-th↗
arXiv2017

A cohomological interpretation of derivations on graded algebras

We trace derivations through Demazure's correspondence between a finitely generated positively graded normal $k$-algebras $A$ and normal projective $k$-varieties $X$ equipped with an ample $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $D$. We obtain a generalized Euler sequence involving a sheaf on $X$ whose space of global sections consists of all homogeneous $k$-linear derivations of $A$ and a sheaf of logarithmic derivations on $X$.

math.AG↗
arXiv2016

On Saito's normal crossing condition

Kyoji Saito defined a residue map from the logarithmic differential 1-forms along a reduced complex analytic hypersurface to the meromorphic functions on the hypersurface. He studied the condition that the image of this map coincides with the weakly holomorphic functions, that is, with the functions on the normalization. With Michel Granger, the author proved that this condition is equivalent to the hypersurface being normal crossing in codimension one. In this article, the condition is given a natural interpretation in terms of regular differential forms beyond the hypersurface case. For reduced equidimensional complex analytic spaces which are free in codimension one, the geometric interpretation of being normal crossing in codimension one is shown to persist.

math.AG↗
arXiv2014

Partial normalizations of Coxeter arrangements and discriminants

We study natural partial normalization spaces of Coxeter arrangements and discriminants and relate their geometry to representation theory. The underlying ring structures arise from Dubrovin's Frobenius manifold structure which is lifted (without unit) to the space of the arrangement. We also describe an independent approach to these structures via duality of maximal Cohen--Macaulay fractional ideals. In the process, we find 3rd order differential relations for the basic invariants of the Coxeter group. Finally, we show that our partial normalizations give rise to new free divisors.

math.AG↗
arXiv2014

Derivations of negative degree on quasihomogeneous isolated complete intersection singularities

J. Wahl conjectured that every quasihomogeneous isolated normal singularity admits a positive grading for which there are no derivations of negative weighted degree. We confirm his conjecture for quasihomogeneous isolated complete intersection singularities of either order at least 3 or embedding dimension at most 5. For each embedding dimension larger than 5 (and each dimension larger than 3), we give a counter-example to Wahl's conjecture.

math.AG↗
arXiv2014

Normal crossing properties of complex hypersurfaces via logarithmic residues

We introduce a dual logarithmic residue map for hypersurface singularities and use it to answer a question of Kyoji Saito. Our result extends a theorem of Lê and Saito by an algebraic characterization of hypersurfaces that are normal crossing in codimension one. For free divisors, we relate the latter condition to other natural conditions involving the Jacobian ideal and the normalization. This leads to an algebraic characterization of normal crossing divisors. As a side result, describe all free divisors with Gorenstein singular locus.

math.AG↗
arXiv2013

Local analysis of Grauert-Remmert-type normalization algorithms

Normalization is a fundamental ring-theoretic operation; geometrically it resolves singularities in codimension one. Existing algorithmic methods for computing the normalization rely on a common recipe: successively enlarge the given ring in form an endomorphism ring of a certain (fractional) ideal until the process becomes stationary. While Vasconcelos' method uses the dual Jacobian ideal, Grauert-Remmert-type algorithms rely on so-called test ideals. For algebraic varieties, one can apply such normalization algorithms globally, locally, or formal analytically at all points of the variety. In this paper, we relate the number of iterations for global Grauert-Remmert-type normalization algorithms to that of its local descendants. We complement our results by an explicit study of ADE singularities. This includes the description of the normalization process in terms of value semigroups of curves. It turns out that the intermediate steps produce only ADE singularities and simple space curve singularities from the list of Fruehbis-Krueger.

math.AG↗
arXiv2013

Adjoint divisors and free divisors

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 plethora of new examples of free divisors. The presented results seem to scratch the surface of a more general phenomenon that is still to be revealed.

math.AG↗
arXiv2012

Local cohomology of logarithmic forms

Let Y be a divisor on a smooth algebraic variety X. We investigate the geometry of the Jacobian scheme of Y, homological invariants derived from logarithmic differential forms along Y, and their relationship with the property that Y is a free divisor. We consider arrangements of hyperplanes as a source of examples and counterexamples. In particular, we make a complete calculation of the local cohomology of logarithmic forms of generic hyperplane arrangements.

math.AG↗
arXiv2012

Resonance equals reducibility for A-hypergeometric systems

Classical theorems of Gel'fand et al., and recent results of Beukers, show that non-confluent Cohen-Macaulay A-hypergeometric systems have reducible monodromy representation if and only if the continuous parameter is A-resonant. We remove both the confluence and Cohen-Macaulayness conditions while simplifying the proof.

math.AG↗
arXiv2012

Energy distribution and quantum yield for photoemission from air-contaminated gold surfaces under UV illumination close to the threshold

The kinetic energy distributions of photo-electrons emitted from gold surfaces under illumination by UV-light close to the threshold are measured and analyzed. Samples are prepared as chemically clean through Ar-Ion sputtering and then exposed to atmosphere for variable durations before Quantum Yield measurements are performed after evacuation. During measurements the bias voltage applied to the sample is varied and the resulting emission current measured. Taking the derivative of the current-voltage curve yields the energy distribution which is found to closely resemble the distribution of total energies derived by DuBridge for emission from a free electron gas. We investigate the dependence of distribution shape and width on electrode geometry and contaminant substances adsorbed from the atmosphere, in particular to water and hydro-carbons. Emission efficiency increases initially during air exposure before diminishing to zero on a timescale of several hours, whilst subsequent annealing of the sample restores emissivity. A model fit function, in good quantitative agreement with the measured data, is introduced which accounts for the experiment-specific electrode geometry and an energy dependent transmission coefficient. The impact of large patch potential fields from contact potential drops between sample and sample holder is investigated. The total quantum yield is split into bulk and surface contributions which are tested for their sensitivity to light incidence angle and polarization. Our results are directly applicable to model parameters for the contact-free discharge system onboard the LISA Pathfinder spacecraft.

cond-mat.mtrl-sci↗
arXiv2012

Freeness and multirestriction of hyperplane arrangements

Generalizing a result of Yoshinaga in dimension 3, we show that a central hyperplane arrangement in 4-space is free exactly if its restriction with multiplicities to a fixed hyperplane of the arrangement is free and its reduced characteristic polynomial equals the characteristic polynomial of this restriction. We show that the same statement holds true in any dimension when imposing certain tameness hypotheses.

math.AG↗
arXiv2011

A geometric deletion-restriction formula

In this paper, we recover the characteristic polynomial of an arrangement of hyperplanes by computing the rational equivalence class of the variety defined by the logarithmic ideal of the arrangement. The logarithmic ideal was introduced in [arXiv:0907.0896v2] in a study of the critical points of the master function. The above result is used to understand the asymptotic behavior the Hilbert series of the logarithmic ideal. As an application, we prove the Solomon-Terao formula under the tame hypothesis by identifying each side of the formula with a certain specialization of the Hilbert series of the logarithmic ideal.

math.AG↗
arXiv2010

Free divisors in prehomogeneous vector spaces

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 Kimura's list of irreducible prehomogeneous vector spaces are the only irreducible reductive linear free divisors. Furthermore, we show that all quiver linear free divisors are strongly Euler homogeneous, that they are locally weakly quasihomogeneous at points whose corresponding representation is not regular, and that all tame quiver linear free divisors are locally weakly quasihomogeneous. In particular, the latter satisfy the logarithmic comparison theorem.

math.AG↗
arXiv2010

Complexes, duality and Chern classes of logarithmic forms along hyperplane arrangements

We describe dualities and complexes of logarithmic forms and differentials for central affine and corresponding projective arrangements. We generalize the Borel-Serre formula from vector bundles to sheaves on projective d-space with locally free resolutions of length one. Combining these results we present a generalization of a formula due to Mustata and Schenck, relating the Poincare polynomial of an arrangement in projective 3-space (or a locally tame arrangement in projective d-space with zero-dimensional non-free locus) to the total Chern polynomial of its sheaf of logarithmic 1-forms.

math.AG↗
arXiv2009

A solvability criterion for the Lie algebra of derivations of a fat point

We consider the Lie algebra of derivations of a zero dimensional local complex algebra. We describe an inequality involving the embedding dimension, the order, and the first deviation that forces this Lie algebra to be solvable. Our result was motivated by and generalizes the solvability of the Yau algebra of an isolated hypersurface singularity.

math.AG↗