arXiv subjects

Daniel Bath

Publications and source records attributed to Daniel Bath.

13 recordsLinked to original sources

arXiv2026

Tropical resolutions of configuration hypersurfaces

Configuration polynomials generalize the Kirchhoff polynomial of a graph, as well as the Symanzik polynomials that appear in the denominators of Feynman integrands. The configuration hypersurfaces cut out by such polynomials are typically highly singular, which poses a challenge for the evaluation of Feynman integrals even in simplified settings. In this paper, we provide a two-step recipe for a resolution of singularities of any irreducible configuration hypersurface. We first consider the normalization of the Nash blow-up, which we identify with an incidence variety introduced by Bloch. This variety is typically still not smooth, but it is the closure of a smooth subvariety of a torus. The latter then a smooth, tropical compactification, using work of Tevelev. We construct explicitly such a compactification and a morphism to the normalized Nash blow-up for every configuration, described in terms of bipermutohedral matroid combinatorics introduced by Ardila, Denham and Huh. Along the way, we find that the normalized Nash blow-up of the configuration hypersurface has strongly $F$-regular singularities in positive characteristic. We deduce this by certifying $F$-rationality of its biprojective cone, and infer from it that the normalized Nash blow-up has rational singularities over the complex numbers.

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arXiv2026

The Strong Monodromy Conjecture for a class of homogeneous polynomials in three variables

We consider the class of all homogeneous, possibly non-reduced, polynomials $f$ whose associated reduced projective divisor $D_{\text{red}} \subset \mathbb{P}^{n-1}$ has (at worst) quasi-homogeneous isolated singularities. In an arbitrary number of variables $n$ and with $d$ denoting the degree of $f$, we characterize when $-n/d$ is a root of the Bernstein--Sato polynomial of $f$ in terms of elementary data involving logarithmic derivations. When we restrict to three variables, we prove the resulting class of polynomials satisfies the Strong Monodromy Conjecture, in the motivic sense.

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arXiv2026

On singularities of determinantal hypersurfaces

Given a closed subscheme $Z$ in a smooth variety $X$, defined by the maximal minors of an $s\times r$ matrix of regular functions, with $s\geq r$, we consider the corresponding incidence correspondence $W$ in $Y=X\times {\mathbf P}^{r-1}$, and relate the log canonical thresholds of $(X,Z)$ and $(Y,W)$. In particular, when $r=s$, we show that ${\rm lct}(X,Z)=1$ if and only if ${\rm lct}(Y,W)=r$. Moreover, in this case, we show that $Z$ has rational singularities if and only if $W$ has pure codimension $r$ in $Y$ and has rational singularities. As a consequence, we deduce that for a configuration hypersurface with a connected configuration matroid, the corresponding configuration incidence variety has rational singularities.

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arXiv2024

Singularities of square-free polynomials

We prove that hypersurfaces defined by irreducible square-free polynomials have rational singularities. As an easy consequence, we deduce that certain (possibly non-square-free) polynomials associated to pairs of square-free polynomials define hypersurfaces with rational singularities. This extends results on certain classes of polynomials associated to matroids and Feynman diagrams in [BW].

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arXiv2024

The Hodge filtration and parametrically prime divisors

We study the canonical Hodge filtration on the sheaf $\mathscr{O}_X(*D)$ of meromorphic functions along a divisor. For a germ of an analytic function $f$ whose Bernstein-Sato's polynomial's roots are contained in $(-2,0)$, we: give a simple algebraic formula for the zeroeth piece of the Hodge filtration; bound the first step of the Hodge filtration containing $f^{-1}$. If we additionally require $f$ to be Euler homogeneous and parametrically prime, then we extend our algebraic formula to compute every piece of the canonical Hodge filtration, proving in turn that the Hodge filtration is contained in the induced order filtration. Finally, we compute the Hodge filtration in many examples and identify several large classes of divisors realizing our theorems.

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arXiv2024

Matroidal polynomials, their singularities, and applications to Feynman diagrams

Given a matroid or flag of matroids we introduce several broad classes of polynomials satisfying Deletion-Contraction identities, and study their singularities. There are three main families of polynomials captured by our approach: matroidal polynomials on a matroid (including matroid basis polynomials, configuration polynomials, Tutte polynomials); flag matroidal polynomials on a flag matroid; and Feynman integrands. The last class includes under general kinematics the inhomogeneous Feynman diagram polynomials which naturally arise in the Lee--Pomeransky form of the Feynman integral attached to a Feynman diagram. Assuming that the primary underlying matroid is connected and of positive rank (and in the flag case, has rank at least two), we show: a) in positive characteristic, homogeneous matroidal polynomials are strongly $F$-regular; b) over an algebraically closed field of characteristic zero, the associated jet schemes of (flag) matroidal polynomials as well as those of Feynman integrands are irreducible. Consequently, all these polynomials have rational singularities (or are smooth).

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arXiv2024

Bernstein--Sato polynomials of locally quasi-homogeneous divisors in $\mathbb{C}^{3}$

We consider the Bernstein--Sato polynomial of a locally quasi-homogeneous polynomial $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$. We construct, in the analytic category, a complex of $\mathscr{D}_{X}[s]$-modules that can be used to compute the $\mathscr{D}_{X}[s]$-dual of $\mathscr{D}_{X}[s] f^{s-1}$ as the middle term of a short exact sequence where the outer terms are well understood. This extends a result by Narv\'{a}ez Macarro where a freeness assumption was required. We derive many results about the zeroes of the Bernstein--Sato polynomial. First, we prove each nonvanishing degree of the zeroeth local cohomology of the Milnor algebra $H_{\mathfrak{m}}^{0} (R / (\partial f))$ contributes a root to the Bernstein--Sato polynomial, generalizing a result of M. Saito's (where the argument cannot weaken homogeneity to quasi-homogeneity). Second, we prove the zeroes of the Bernstein--Sato polynomial admit a partial symmetry about $-1$, extending a result of Narv\'{a}ez Macarro that again required freeness. We give applications to very small roots, the twisted Logarithmic Comparison Theorem, and more precise statements when $f$ is additionally assumed to be homogeneous. Finally, when $f$ defines a hyperplane arrangement in $\mathbb{C}^{3}$ we give a complete formula for the zeroes of the Bernstein--Sato polynomial of $f$. We show all zeroes except the candidate root $-2 + (2 / \text{deg}(f))$ are (easily) combinatorially given; we give many equivalent characterizations of when the only non-combinatorial candidate root $-2 + (2/ \text{deg}(f))$ is in fact a zero of the Bernstein--Sato polynomial. One equivalent condition is the nonvanishing of $H_{\mathfrak{m}}^{0}( R / (\partial f))_{\text{deg}(f) - 1}$.

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arXiv2022

Twisted logarithmic complexes of positively weighted homogeneous divisors

For a rank 1 local system on the complement of a reduced divisor on a complex manifold $X$, its cohomology is calculated by the twisted meromorphic de Rham complex. Assuming the divisor is everywhere positively weighted homogeneous, we study necessary or sufficient conditions for a quasi-isomorphism from its twisted logarithmic subcomplex, called the logarithmic comparison theorem (LCT), by using a stronger version in terms of the associated complex of $D_X$-modules. In case the connection is a pullback by a defining function $f$ of the divisor and the residue is $\alpha$, we prove among others that if LCT holds, the annihilator of $f^{\alpha-1}$ in $D_X$ is generated by first order differential operators and $\alpha-1-j$ is not a root of the Bernstein-Sato polynomial for any positive integer $j$. The converse holds assuming either of the two conditions in case the associated complex of $D_X$-modules is acyclic except for the top degree. In the case where the local system is constant, the divisor is defined by a homogeneous polynomial, and the associated projective hypersurface has only weighted homogeneous isolated singularities, we show that LCT is equivalent to that $-1$ is the unique integral root of the Bernstein-Sato polynomial. We also give a simple proof of LCT in the hyperplane arrangement case under appropriate assumptions on residues, which is an immediate corollary of higher cohomology vanishing associated with Castelnuovo-Mumford regularity. Here the zero-extension case is also treated.

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arXiv2022

Hyperplane Arrangements Satisfy (un)Twisted Logarithmic Comparison Theorems, Applications to $\mathscr{D}_{X}$-modules

For a reduced hyperplane arrangement we prove the analytic Twisted Logarithmic Comparison Theorem, subject to mild combinatorial arithmetic conditions on the weights defining the twist. This gives a quasi-isomorphism between the twisted logarithmic de Rham complex and the twisted meromorphic de Rham complex. The latter computes the cohomology of the arrangement's complement with coefficients from the corresponding rank one local system. We also prove the algebraic variant (when the arrangement is central), and the analytic and algebraic (untwisted) Logarithmic Comparison Theorems. The last item positively resolves an old conjecture of Terao. We also prove that: every nontrivial rank one local system on the complement can be computed via these Twisted Logarithmic Comparison Theorems; these computations are explicit finite dimensional linear algebra. Finally, we give some $\mathscr{D}_{X}$-module applications: for example, we give a sharp restriction on the codimension one components of the multivariate Bernstein--Sato ideal attached to an arbitrary factorization of an arrangement. The bound corresponds to (and, in the univariate case, gives an independent proof of) M. Saito's result that the roots of the Bernstein--Sato polynomial of a non-smooth, central, reduced arrangement live in $(-2 + 1/d, 0).$

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arXiv2021

A noncommutative analogue of the Peskine--Szpiro Acyclicity Lemma

We present a variant of the Peskine--Szpiro Acyclicity Lemma, and hence a way to certify exactness of a complex of finite modules over a large class of (possibly) noncommutative rings. Specifically, over the class of Auslander regular rings. In the case of relative $\mathscr{D}_{X}$-modules, for example $\mathscr{D}_{X}[s_{1}, \dots, s_{r}]$-modules, the hypotheses have geometric realizations making them easier to authenticate. We demonstrate the efficacy of this lemma and its various forms by: independently recovering some results related to Bernstein--Sato polynomials; establishing a new result about quasi-free structures of free multi-derivations of hyperplane arrangements.

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arXiv2020

A Note on Bernstein-Sato Varieties for Tame Divisors and Arrangements

For strongly Euler-homogeneous, Saito-holonomic, and tame analytic germs we consider general types of multivariate Bernstein-Sato ideals associated to arbitrary factorizations of our germ. We show the zero loci of these ideals are purely codimension one and the zero loci associated to different factorizations are related by a diagonal property. If, additionally, the divisor is a hyperplane arrangement, we show the Bernstein-Sato ideals attached to a factorization into linear forms are principal. As an application, we independently verify and improve an estimate of Maisonobe's regarding standard Bernstein-Sato ideals for reduced, generic arrangements: we compute the Bernstein-Sato ideal for a factorization into linear forms and we compute its zero locus for other factorizations.

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arXiv2019

Combinatorially Determined Zeroes of Bernstein--Sato Ideals for Tame and Free Arrangements

For a central, not necessarily reduced, hyperplane arrangement $f$ equipped with any factorization $f = f_{1} \cdots f_{r}$ and for $f^{\prime}$ dividing $f$, we consider a more general type of Bernstein--Sato ideal consisting of the polynomials $B(S) \in \mathbb{C}[s_{1}, \dots, s_{r}]$ satisfying the functional equation $B(S) f^{\prime} f_{1}^{s_{1}} \cdots f_{r}^{s_{r}} \in \text{A}_{n}(\mathbb{C})[s_{1}, \dots, s_{r}] f_{1}^{s_{1} + 1} \cdots f_{r}^{s_{r} + 1}.$ Generalizing techniques due to Maisonobe, we compute the zero locus of the standard Bernstein--Sato ideal in the sense of Budur (i.e. $f^{\prime} = 1)$ for any factorization of a free and reduced $f$ and for certain factorizations of a non-reduced $f$. We also compute the roots of the Bernstein--Sato polynomial for any power of a free and reduced arrangement. If $f$ is tame, we give a combinatorial formula for the roots lying in $[-1,0).$ For $f^{\prime} \neq 1$ and any factorization of a line arrangement, we compute the zero locus of this ideal. For free and reduced arrangements of larger rank, we compute the zero locus provided $\text{deg}(f^{\prime}) \leq 4$ and give good estimates otherwise. Along the way we generalize a duality formula for $\mathscr{D}_{X,\mathfrak{x}}[S]f^{\prime}f_{1}^{s_{1}} \cdots f_{r}^{s_{r}}$ that was first proved by Narv\'aez-Macarro for $f$ reduced, $f^{\prime} = 1$, and $r = 1.$ As an application, we investigate the minimum number of hyperplanes one must add to a tame $f$ so that the resulting arrangement is free. This notion of freeing a divisor has been explicitly studied by Mond and Schulze, albeit not for hyperplane arrangements. We show that small roots of the Bernstein--Sato polynomial of $f$ can force lower bounds for this number.

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arXiv2019

Bernstein-Sato Varieties and Annihilation of Powers

Given a complex germ $f$ near the point $\mathfrak{x}$ of the complex manifold $X$, equipped with a factorization $f = f_{1} \cdots f_{r}$, we consider the $\mathscr{D}_{X,\mathfrak{x}}[s_{1}, \dots, s_{r}]$-module generated by $ F^{S} := f_{1}^{s_{1}} \cdots f_{r}^{s_{r}}$. We show for a large class of germs that the annihilator of $F^{S}$ is generated by derivations and this property does not depend on the chosen factorization of $f$. We further study the relationship between the Bernstein-Sato variety attached to $F$ and the cohomology support loci of $f$, via the $\mathscr{D}_{X,\mathfrak{x}}$-map $\nabla_{A}$. This is related to multiplication by $f$ on certain quotient modules. We show that for our class of divisors the injectivity of $\nabla_{A}$ implies its surjectivity. Restricting to reduced, free divisors, we also show the reverse, using the theory of Lie-Rinehart algebras. In particular, we analyze the dual of $\nabla_{A}$ using techniques pioneered by Narv\'aez-Macarro. As an application of our results we establish a conjecture of Budur in the tame case: if $\text{V}(f)$ is a central, essential, indecomposable, and tame hyperplane arrangement, then the Bernstein-Sato variety associated to $F$ contains a certain hyperplane. By the work of Budur, this verifies the Topological Mulivariable Strong Monodromy Conjecture for tame arrangements. Finally, in the reduced and free case, we characterize local systems outside the cohomology support loci of $f$ near $\mathfrak{x}$ in terms of the simplicity of modules derived from $F^{S}.$

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