arXiv subjects

Andreas Kretschmer

Publications and source records attributed to Andreas Kretschmer.

13 recordsLinked to original sources

arXiv2026

On the Brill--Noether Theory of Sheaves on Regular Surfaces

We study the Brill--Noether theory of higher rank sheaves on surfaces $X$ with $H^1(X,\mathcal{O}_X) = 0$ and its interactions with the classical Brill--Noether theory of line bundles on curves lying on $X$. Denoting by $\mathrm{Spl}(v)$ the moduli space of simple sheaves on $X$ with Chern character $v = (r,c_1,\mathrm{ch}_2)$, we introduce the Brill--Noether loci $BN^k(v)^\circ \subseteq \mathrm{Spl}(v)$ of \emph{generically $k$-generated} sheaves and show, under assumptions on $c_1$ and $K_X$, certain well-behavedness results. In particular, if $c_1$ is \emph{indecomposable} and $X$ is a K3 surface, we generalize a Brill--Noether theorem for sheaves on K3s due to Yoshioka. This is achieved by employing He's deformation theory of the moduli space of coherent systems and a globalization of the Lazarsfeld--Mukai construction, providing a correspondence between the Brill--Noether theories of generically $r$-generated sheaves on $X$ and that of line bundles on curves $C \in |c_1|$. The same techniques yield upper bounds on the dimension of $\mathcal{W}^{r,\mathrm{bpf}}_d(|c_1|) \setminus \mathcal{W}^{r+1}_d(|c_1|)$ consisting of basepoint-free line bundles in terms of the number of endomorphisms of associated Lazarsfeld--Mukai bundles, for arbitrary $c_1$. This generalizes a result of Aprodu and Farkas in the context of Green's conjecture beyond K3 surfaces. In the case $r = 1$ of pencils, we obtain smoothness and expected-dimension results under an effectivity condition on $-K_X|_C$ for $C \in |c_1|$.

math.AG↗

arXiv2026

From Local Structure to Thermodynamics and Transport of Water with Machine Learning Force Fields

We evaluate machine learning force fields derived from different density functional theory exchange correlation functionals using the full six-dimensional pair correlation function of liquid water, three-body structural descriptors, excess entropy, and transport properties. The predicted microscopic structure and dynamics depend strongly on the underlying functional: neglecting dispersion produces pronounced overstructuring, overly negative excess entropy, and suppressed diffusion. Translational and orientational entropy contributions are tightly coupled and together exhibit a clear relationship with the reduced selfdiffusion coefficient. Among the tested models, RPBE-D3 provides the most consistent agreement with experiment across structural, thermodynamic, and transport properties. The classical SPC/E model serves as an additional reference and displays notable similarities to RPBE-D3, consistent with the comparable Born effective and partial charges of the two models.

cond-mat.soft↗

arXiv2026

Inverse Eigenvalue Problems, Floquet Isospectrality and the Hilbert--Chow Morphism

When can one change the diagonal of a matrix without changing its spectrum? We completely answer this question over an algebraically closed field of characteristic zero or larger than the size of the matrix: An $n \times n$ matrix $A$ admits a nonzero diagonal matrix $D$ such that $A$ and $A+D$ have the same spectrum if and only if, for some size $k$, the $k \times k$ principal minors of $A$ are not all equal. This relates to the classical additive inverse eigenvalue problem in numerical analysis and has implications for existence and rigidity results in the theory of Floquet isospectrality of discrete periodic operators in solid state physics. The proof employs new techniques involving Hilbert schemes of points and the infinitesimal structure of the Hilbert--Chow morphism.

math.AG↗

arXiv2024

Logarithmic Discriminants of Hyperplane Arrangements

A recurring task in particle physics and statistics is to compute the complex critical points of a product of powers of affine-linear functions. The logarithmic discriminant characterizes exponents for which such a function has a degenerate critical point in the corresponding hyperplane arrangement complement. We study properties of this discriminant, exploiting its connection with the Hurwitz form of a reciprocal linear space.

math.AG↗

arXiv2024

Symmetric Ideals and Invariant Hilbert Schemes

A symmetric ideal is an ideal in a polynomial ring which is stable under all permutations of the variables. In this paper we initiate a global study of zero-dimensional symmetric ideals. By this we mean a geometric study of the invariant Hilbert schemes $\mathrm{Hilb}_{\rho}^{S_n}(\mathbb{C}^n)$ parametrizing symmetric subschemes of $\mathbb{C}^n$ whose coordinate rings, as $S_n$-modules, are isomorphic to a given representation $\rho$. In the case that $\rho = M^\lambda$ is a permutation module corresponding to certain special types of partitions $\lambda$ of $n$, we prove that $\mathrm{Hilb}_{\rho}^{S_n}(\mathbb{C}^n)$ is irreducible or even smooth. We also prove irreducibility whenever $\dim \rho \leq 2n$ and the invariant Hilbert scheme is non-empty. In this same range, we classify all homogeneous symmetric ideals and decide which of these define singular points of $\mathrm{Hilb}_{\rho}^{S_n}(\mathbb{C}^n)$. A central tool is the combinatorial theory of higher Specht polynomials.

math.AG↗

arXiv2023

Thin polytopes: Lattice polytopes with vanishing local $h^*$-polynomial

In this paper we study the novel notion of thin polytopes: lattice polytopes whose local $h^*$-polynomials vanish. The local $h^*$-polynomial is an important invariant in modern Ehrhart theory. Its definition goes back to Stanley with fundamental results achieved by Karu, Borisov & Mavlyutov, Schepers, and Katz & Stapledon. The study of thin simplices was originally proposed by Gelfand, Kapranov and Zelevinsky, where in this case the local $h^*$-polynomial simply equals its so-called box polynomial. Our main results are the complete classification of thin polytopes up to dimension 3 and the characterization of thinness for Gorenstein polytopes. The paper also includes an introduction to the local $h^*$-polynomial with a survey of previous results.

math.CO↗

arXiv2023

Explaining the Entropy Forming Ability with the atomic size mismatch

To quickly screen for single-phased multi-principal-element materials, a so-called entropy forming ability (EFA) parameter is sometimes used as a descriptor. The higher the EFA, the higher is the propensity to form a single-phase structure, which is stabilized against separation up to a certain threshold by the configurational entropy. We have investigated this EFA descriptor with atomic relaxations in special-quasi-random structures and discovered that the EFA correlates inversely with the lattice distortion. Large atomic size differences lead to multi-phase compounds, and little size differences to single-phase compounds. Instead of configurational entropy, we therefore demonstrate the applicability of the Hume-Rothery rules to phase stability of solid solutions even in compositionally complex ceramics.

cond-mat.mtrl-sci↗

arXiv2022

Ideals of submaximal minors of sparse symmetric matrices

We study algebraic and homological properties of the ideal of submaximal minors of a sparse generic symmetric matrix. This ideal is generated by all $(n-1)$-minors of a symmetric $n \times n$ matrix whose entries in the upper triangle are distinct variables or zeros and the zeros are only allowed at off-diagonal places. The surviving off-diagonal entries are encoded as a simple graph $G$ with $n$ vertices. We prove that the minimal free resolution of this ideal is obtained from the case without any zeros via a simple pruning procedure, extending methods of Boocher. This allows us to compute all graded Betti numbers in terms of $n$ and a single invariant of $G$. Moreover, it turns out that these ideals are always radical and have Cohen--Macaulay quotients if and only if $G$ is either connected or has no edges at all. The key input are some new Gröbner basis results with respect to non-diagonal term orders associated to $G$.

math.AC↗

arXiv2022

When are symmetric ideals monomial?

We study conditions on polynomials such that the ideal generated by their orbits under the symmetric group action becomes a monomial ideal or has a monomial radical. If the polynomials are homogeneous, we expect that such an ideal has a monomial radical if their coefficients are sufficiently general with respect to their supports. We prove this for instance in the case where some generator contains a power of a variable. Moreover, if the polynomials have only square-free terms and their coefficients do not sum to zero, then in a larger polynomial ring the ideal itself is square-free monomial. This has implications also for symmetric ideals of the infinite polynomial ring.

math.AC↗

arXiv2022

The enumerative geometry of cubic hypersurfaces: point and line conditions

In order to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points, we construct a compactification of their moduli space. We term the latter a $1$--\textit{complete variety of cubic hypersurfaces} in analogy to the space of complete quadrics. Paolo Aluffi explored the case of plane cubic curves. Starting from his work, we construct such a space in arbitrary dimension by a sequence of five blow-ups. The counting problem is then reduced to the computation of five Chern classes, climbing the sequence of blow-ups. Computing the last of these is difficult due to the fact that the vector bundle is not given explicitly. Identifying a restriction of this vector bundle, we arrive at the desired numbers in the case of cubic surfaces.

math.AG↗

arXiv2021

The geometry of Gaussian double Markovian distributions

Gaussian double Markovian models consist of covariance matrices constrained by a pair of graphs specifying zeros simultaneously in the covariance matrix and its inverse. We study the semi-algebraic geometry of these models, in particular their dimension, smoothness and connectedness as well as algebraic and combinatorial properties.

math.ST↗

arXiv2021

Liftings of polynomial systems decreasing the mixed volume

The BKK theorem states that the mixed volume of the Newton polytopes of a system of polynomial equations upper bounds the number of isolated torus solutions of the system. Homotopy continuation solvers make use of this fact to pick efficient start systems. For systems where the mixed volume bound is not attained, such methods are still tracking more paths than necessary. We propose a strategy of improvement by lifting a system to an equivalent system with a strictly lower mixed volume at the expense of more variables. We illustrate this idea providing lifting constructions for arbitrary bivariate systems and certain dense-enough systems.

math.AG↗

arXiv2020

The Chow ring of hyperkähler varieties of $K3^{[2]}$-type via Lefschetz actions

We propose an explicit conjectural lift of the Neron-Severi Lie algebra of a hyperkähler variety $X$ of $K3^{[2]}$-type to the Chow ring of correspondences ${\rm CH}^\ast(X \times X)$ in terms of a canonical lift of the Beauville-Bogomolov class obtained by Markman. We give evidence for this conjecture in the case of the Hilbert scheme of two points of a $K3$ surface and in the case of the Fano variety of lines of a very general cubic fourfold. Moreover, we show that the Fourier decomposition of the Chow ring of $X$ of Shen and Vial agrees with the eigenspace decomposition of a canonical lift of the grading operator.

math.AG↗