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A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, the mixed characteristic minimal model program, Cohen-Macaulayness of absolute integral closures and cohomology vanishing theorems. Nevertheless, many basic questions about these singularities remain elusive. One outstanding problem is whether the splinter property spreads from a point to an open neighborhood of a noetherian scheme. Our paper addresses this problem in prime characteristic, where we show that a locally noetherian scheme that has finite Frobenius or that is locally essentially of finite type over a quasi-excellent local ring has an open splinter locus. In particular, all varieties over fields of positive characteristic have open splinter loci. Intimate connections are established between the openness of splinter loci and $F$-compatible ideals, which are prime characteristic analogues of log canonical centers. We prove the surprising fact that for a large class of noetherian rings with pure (aka universally injective) Frobenius, the splinter condition is detected by the splitting of a single generically \'etale finite extension. We also show that for a noetherian $\textbf{N}$-graded ring over a field, the homogeneous maximal ideal detects the splinter property., Comment: Comments welcome, minor updates in v2

In this paper, we consider the homogeneous coordinate rings $A(Y_{n,d}) \cong \mathbb{K}[\Omega_{n,d}]$ of monomial projections $Y_{n,d}$ of Veronese varieties parameterized by subsets $\Omega_{n,d}$ of monomials of degree $d$ in $n+1$ variables where: (1) $\Omega_{n,d}$ contains all monomials supported in at most $s$ variables and, (2) $\Omega_{n,d}$ is a set of monomial invariants of a finite diagonal abelian group $G \subset GL(n+1,\mathbb{K})$ of order $d$. Our goal is to study when $\mathbb{K}[\Omega_{n,d}]$ is a quadratic algebra and, if so, when $\mathbb{K}[\Omega_{n,d}]$ is Koszul or G-quadratic. For the family (1), we prove that $\mathbb{K}[\Omega_{n,d}]$ is quadratic when $s \ge \lceil \frac{n+2}{2} \rceil$. For the family (2), we completely characterize when $\mathbb{K}[\Omega_{2,d}]$ is quadratic in terms of the group $G \subset GL(3,\mathbb{K})$, and we prove that $\mathbb{K}[\Omega_{2,d}]$ is quadratic if and only if it is Koszul. We also provide large families of examples where $\mathbb{K}[\Omega_{n,d}]$ is G-quadratic., Comment: To appear in Journal of Algebra

Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra $A$ having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is greater than the socle degree of $A$, we show that $A$ is almost strong Lefschetz. This quite modest result is nevertheless arguably the most encompassing so far concerning the strong Lefschetz property for graded codimension three AG algebras.

We show that a sufficiently general hypersurface of degree $d$ in $n$-dimensional projective space admits a toric Gr\"obner degeneration if and only if $d\leq 2n-1$., Comment: 6 pages; v2 minor revisions. To appear in Canadian Mathematical Bulletin

Let $ e_1, ..., e_c $ be positive integers and let $ Y \subseteq \mathbb{P}^n$ be the monomial complete intersection defined by the vanishing of $x_1^{e_1}, ..., x_c^{e_c}$. In this paper we study sharp upper bounds on the number of equations and syzygies of subschemes parametrized by the Hilbert scheme of points $Hilb^d(Y)$, and discuss applications to the Hilbert scheme of points $Hilb^d(X)$ of arbitrary complete intersections $X \subseteq \mathbb{P}^n$., Final version

We study exceptional minuscule Schubert varieties and provide the defining equations of the defining ideals of their intersection with the big open cell. We also provide the resolutions of these ideals and characterize some of them in terms of fundamental examples of ideals in the theory of Gorenstein ideals., Comment: 21 pages, comments welcome

Let $(R,\mathfrak{m}, k)$ be a strictly local normal $k$-domain of positive characteristic and $P$ be a prime divisor on $X=\text{Spec } R$. We study the Galois category of finite covers over $X$ that are at worst tamely ramified over $P$ in the sense of Grothendieck--Murre. Assuming that $(X,P)$ is a purely $F$-regular pair, our main result is that every Galois cover $f \: Y \to X$ in that Galois category satisfies that $\bigl(f^{-1}(P)\bigr)_{\text{red}}$ is a prime divisor. We shall explain why this should be thought as a (partial) generalization of a classical theorem due to S.S.~Abhyankar regarding the \'etale-local structure of tamely ramified covers between normal schemes with respect to a divisor with normal crossings. Additionally, we investigate the formal consequences this result has on the structure of the fundamental group representing the Galois category. We also obtain a characteristic zero analog by reduction to positive characteristics following Bhatt--Gabber--Olsson's methods., Comment: 45 pages, comments are welcome, typos fixed, shortened down, some parts were rewritten to improve exposition, Proposition 4.8 was removed as it was flawed v3: Major revision, to apper in ANT

We characterize Hilbert polynomials that give rise to Hilbert schemes with two Borel-fixed points and determine when the associated Hilbert schemes or their irreducible components are smooth. In particular, we show that the Hilbert scheme is reduced and has at most two irreducible components. By describing the singularities in a neighbourhood of the Borel-fixed points, we prove that the irreducible components are Cohen-Macaulay and normal. We end by giving many examples of Hilbert schemes with three Borel-fixed points., Comment: To appear in Journal of Algebra

We show that the hessian map of quartic plane curves is a birational morphism onto its image, thus bringing new evidence for a very interesting conjecture of Ciro Ciliberto and Giorgio Ottaviani. Our new approach also yields a simpler proof of the similar property for cubic surfaces, which is already known by the work of these two authors.

Let $C:f=0$ be a reduced curve in the complex projective plane. The minimal degree $mdr(f)$ of a Jacobian syzygy for $f$, which is the same as the minimal degree of a derivation killing $f$, is an important invariant of the curve $C$, for instance it can be used to determined whether $C$ is free or nearly free. In this note we study the relations of this invariant $mdr(f)$ with a decomposition of $C$ as a union of two curves $C_1$ and $C_2$, without common irreducible components. When all the singularities that occur are quasihomogeneous, a result by Schenck, Terao and Yoshinaga yields finer information on this invariant in this setting. Using this, we give some geometrical criteria, the first ones of this type in the existing literature as far as we know, for a line to be a jumping line for the rank 2 vector bundle of logarithmic vector fields along a reduced curve $C$., Comment: v4. Proposition 5.5 modified, one reference updated, the rest unchanged

Given a generically finite local extension of valuation rings $V \subset W$, the question of whether $W$ is the localization of a finitely generated $V$-algebra is significant for approaches to the problem of local uniformization of valuations using ramification theory. Hagen Knaf proposed a characterization of when $W$ is essentially of finite type over $V$ in terms of classical invariants of the extension of associated valuations. Knaf's conjecture has been verified in important special cases by Cutkosky and Novacoski using local uniformization of Abhyankar valuations and resolution of singularities of excellent surfaces in arbitrary characteristic, and by Cutkosky for valuation rings of function fields of characteristic $0$ using embedded resolution of singularities. In this paper we prove Knaf's conjecture in full generality., Comments very welcome!

We prove a Galois-type correspondence between compositions of purely inseparable field extensions (including infinite ones) and subalgebras of differential operators. This correspondence can be utilized to establish a connection between separable field extensions and purely inseparable field extensions. Specifically, it serves as a progression towards gaining a deeper comprehension of a foliation theory on varieties in positive characteristic., 29 pages. Comments welcome!

We survey some recent development in the stability theory of klt singularities. The main focus is on the solution of the stable degeneration conjecture., Submission to Handbook of Singularities. 42 pages. Comments are welcome

Given an ideal $I$ in a polynomial ring $K[x_1,\dots,x_n]$ over a field $K$, we present a complete algorithm to compute the binomial part of $I$, i.e., the subideal ${\rm Bin}(I)$ of $I$ generated by all monomials and binomials in $I$. This is achieved step-by-step. First we collect and extend several algorithms for computing exponent lattices in different kinds of fields. Then we generalize them to compute exponent lattices of units in 0-dimensional $K$-algebras, where we have to generalize the computation of the separable part of an algebra to non-perfect fields in characteristic $p$. Next we examine the computation of unit lattices in affine $K$-algebras, as well as their associated characters and lattice ideals. This allows us to calculate ${\rm Bin}(I)$ when $I$ is saturated with respect to the indeterminates by reducing the task to the 0-dimensional case. Finally, we treat the computation of ${\rm Bin}(I)$ for general ideals by computing their cellular decomposition and dealing with finitely many special ideals called $(s,t)$-binomial parts. All algorithms have been implemented in SageMath., 29 pages

We review the open problems in the theory of deformations of zero-dimensional objects, such as algebras, modules or tensors. We list both the well-known ones and some new ones that emerge from applications. In view of many advances in recent years, we can hope that all of them are in the range of current methods.

We investigate the injectivity of the Frobenius map on thickenings of smooth varieties in projective space over a field of positive characteristic. We obtain uniform bounds -- i.e., independent of the characteristic -- on the thickening that ensures an injective Frobenius map when the projective variety is a smooth complete intersection or an arbitrary projective embedding of an elliptic curve. Our bounds are sharp in the case of hypersurfaces, and in the case of elliptic curves., Comments welcome!

The purpose of this paper is to prove that one can read off the gonality sequence of a smooth projective curve from syzygies of secant varieties of the curve embedded by a line bundle of sufficiently large degree. More precisely, together with Ein-Niu-Park's theorem, our main result shows that the gonality sequence of a smooth projective curve completely determines the shape of the minimal free resolutions of secant varieties of the curve of sufficiently large degree. This is a natural generalization of the gonality conjecture on syzygies of smooth projective curves established by Ein-Lazarsfeld and Rathmann to the secant varieties., 22 pages, any comments are welcome

This paper describes the RationalMaps package for Macaulay2. This package provides functionality for computing several aspects of rational maps such as whether a map is birational, or a closed embedding., Comment: 9 pages. The current version of the package (and other necessary files, such as the latest version of FastMinors.m2) can be accessed at https://github.com/Macaulay2/Workshop-2016-Utah/tree/master/RationalMaps

We introduce a method for associating a chain complex to a module over a combinatorial category, such that if the complex is exact then the module has a rational Hilbert series. We prove homology--vanishing theorems for these complexes for several combinatorial categories including: the category of finite sets and injections, the opposite of the category of finite sets and surjections, and the category of finite dimensional vector spaces over a finite field and injections. Our main applications are to modules over the opposite of the category of finite sets and surjections, known as $FS^{op}$ modules. We obtain many constraints on the sequence of symmetric group representations underlying a finitely generated $FS^{op}$ module. In particular, we describe its character in terms of functions that we call character exponentials. Our results have new consequences for the character of the homology of the moduli space of stable marked curves, and for the equivariant Kazhdan-Luzstig polynomial of the braid matroid., Comment: Corrections to definitions in Section 9, to appear in Alg. & Number Thy

We show that if X_n is a variety of cxn-matrices that is stable under the group Sym([n]) of column permutations and if forgetting the last column maps X_n into X_{n-1}, then the number of Sym([n])-orbits on irreducible components of X_n is a quasipolynomial in n for all sufficiently large n. To this end, we introduce the category of affine FI^op-schemes of width one, review existing literature on such schemes, and establish several new structural results about them. In particular, we show that under a shift and a localisation, any width-one FI^op-scheme becomes of product form, where X_n=Y^n for some scheme Y in affine c-space. Furthermore, to any FI^op-scheme of width one we associate a component functor from the category FI of finite sets with injections to the category PF of finite sets with partially defined maps. We present a combinatorial model for these functors and use this model to prove that Sym([n])-orbits of components of X_n, for all n, correspond bijectively to orbits of a groupoid acting on the integral points in certain rational polyhedral cones. Using the orbit-counting lemma for groupoids and theorems on quasipolynomiality of lattice point counts, this yields our Main Theorem., final version, 40 pages, added several examples and clarifications and corrected typos---with many thanks to a referee!

International audience; We address the description of the tropicalization of families of rational varieties under parametrizations with prescribed support, via curve valuations. We recover and extend results by Sturmfels, Tevelev and Yu for generic coefficients, considering rational parametrizations with non-trivial denominator. The advantage of our point of view is that it can be generalized to deal with non-generic parametrizations. We provide a detailed analysis of the degree of the closed image, based on combinatorial conditions on the relative positions of the supports of the polynomials defining the parametrization. We obtain a new formula and finer bounds on the degree, when the supports of the polynomials are different. We also present a new formula and bounds for the order at the origin in case the closed image is a hypersurface.

We prove that if $f$ is a reduced homogenous polynomial of degree $d$, then its $F$-pure threshold at the unique homogeneous maximal ideal is at least $\frac{1}{d-1}$. We show, furthermore, that its $F$-pure threshold equals $\frac{1}{d-1}$ if and only if $f\in \mathfrak m^{[q]}$ and $d=q+1$, where $q$ is a power of $p$. Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such "extremal singularities," and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are "extremal," for example, in terms of the configurations of lines they can contain., to appear in Transactions of the AMS

This paper considers the problems of finite determinacy and approximation of flat analytic maps from germs of real or complex analytic spaces. It is shown that the flatness of analytic maps from germs of real or complex analytic spaces whose local rings are Cohen-Macaulay is finitely determined. Further, it is shown that flat maps from complete intersection and Cohen-Macaulay analytic germs can be arbitrarily closely approximated by algebraic and Nash maps respectively in such a way that the Hilbert-Samuel function of the special fibre is preserved. It is also proved that in the complex case the preservation of the Hilbert-Samuel function implies the preservation of Whitney's tangent cone., Comment: 13 pages. Substantial improvements in writing, existing theorems strengthened, several results added. arXiv admin note: text overlap with arXiv:1910.11498

We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields of arbitrary characteristic. Furthermore, from the same perspective, we study a family of cluster algebras which are not of finite type and which arise from a star shaped quiver., Comment: 37 pages, minor changes, accepted for publication in Glasgow Mathematical Journal

In this paper we give a complete answer to a question posed by Dimca and Greuel about the quotient of the Milnor and Tjurina numbers of a plane curve singularity. We put this question into a general framework of the study of the difference of Milnor and Tjurina numbers for isolated complete intersection singularities showing its connection with other problems in singularity theory., To appear in Math. Nachr.; 11 pages, Improved redaction of section 3.1., minor changes

In this text, we wish to provide the reader with a short guide to recent works on the theory of dilatations in Commutative Algebra and Algebraic Geometry. These works fall naturally into two categories: one emphasises foundational and theoretical aspects and the other applications to existing theories., arXiv admin note: text overlap with arXiv:2303.07712

In this paper we show that any $\mathbb{A}^2$-fibration over a discrete valuation ring which is also an $\mathbb{A}^2$-form is necessarily a polynomial ring. Further we show that separable $\mathbb{A}^2$-forms over PIDs are trivial.

Let $F$ be a field, let $D$ be a local subring of $F$, and let Val$_F(D)$ be the space of valuation rings of $F$ that dominate $D$. We lift Zariski's connectedness theorem for fibers of a projective morphism to the Zariski-Riemann space of valuation rings of $F$ by proving that a subring $R$ of $F$ dominating $D$ is local, residually algebraic over $D$ and integrally closed in $F$ if and only if there is a closed and connected subspace $Z$ of Val$_F(D)$ such that $R$ is the intersection of the rings in $Z$. Consequently, the intersection of the rings in any closed and connected subset of Val$_F(D)$ is a local ring. In proving this, we also prove a converse to Zariski's connectedness theorem. Our results do not require the rings involved to be Noetherian., 14 pages. Comments welcome!

We prove an effective stabilization result for the sheaf cohomology groups of line bundles on flag varieties parametrizing complete flags in k^n, as well as for the sheaf cohomology groups of polynomial functors applied to the cotangent sheaf Omega on projective space. In characteristic zero, these are natural consequences of the Borel-Weil-Bott theorem, but in characteristic p>0 they are non-trivial. Unlike many important contexts in modular representation theory, where the prime characteristic p is assumed to be large relative to n, in our study we fix p and let n go to infinity. We illustrate the general theory by providing explicit stable cohomology calculations in a number of cases of interest. Our examples yield cohomology groups where the number of indecomposable summands has super-polynomial growth, and also show that the cohomological degrees where non-vanishing occurs do not form a connected interval. In the case of polynomial functors of Omega, we prove a Kunneth formula for stable cohomology, and show the invariance of stable cohomology under Frobenius, which combined with the Steinberg tensor product theorem yields calculations of stable cohomology for an interesting class of simple polynomial functors arising in the work of Doty. The results in the special case of symmetric powers of Omega provide a nice application to commutative algebra, yielding a sharp vanishing result for Koszul modules of finite length in all characteristics.

Let $X$ be a quasi projective scheme over a noetherian affine scheme $Spec(A)$, $U\subseteq X$ be an open subset, and $Z=X-U$. Consider the map $q:{\mathbb K}\left({\mathscr V}(X)\right) \rightarrow {\mathbb K}\left({\mathscr V} U)\right)$ of the ${\mathbb K}$-theory spectra. We give a description of the homotopy fiber of $q$. Let $C{\mathbb M}^Z\left({\mathscr V}(X) \right)$ denote the full subcategory of ${\mathscr F} \in Coh(X)$ such that (1) ${\mathscr F}_{|U}=0$, (2) $grade(Z, X)=grade({\mathscr F} )=\dim_{{\mathscr V}(X)}{\mathscr F} $. It turns out that the homotopy fiber of $q$ is the ${\mathbb K}$-theory spectra ${\mathbb K}\left(C{\mathbb M}^Z\left({\mathscr V}(X) \right)\right)$. Likewise, we compute the homotopy fiber of the the pullback map $$g: {\mathbb G}W\left({\mathscr V}(X)\right) \rightarrow {\mathbb G}W\left({\mathscr V}(U)\right)$$ of Grothendieck-Witt bispectra. Consequently, we obtain long exact sequences of ${\mathbb K}$-groups and of ${\mathbb G}W$-groups., Send comments

In this paper we generalize some results by Siersma, Pellikaan, and de Jong regarding morsifications of singular hypersurfaces whose singular locus is a smooth curve, and present some applications to the study of Yomdin-type isolated singularities. In order to prove these results, we discuss the transversal discriminant of such singularities and how it relates to other algebraic and topological invariants., 29 pages, 2 tables

An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and related fields., 15 pages

In this article we show that the symbolic Rees algebra of a mixed ladder determinantal ideal is strongly $F$-regular. Furthermore, we prove that the symbolic associated graded algebra of a mixed ladder determinantal ideal is $F$-pure. The latter implies that mixed ladder determinantal rings are $F$-pure. We also show that ideals of the poset of minors of a generic matrix give rise to $F$-pure algebras with straightening law., 19 pages, 2 figures

We study the algebraic sets of pairs of matrices defined by the vanishing of the anti-diagonal as well as the cross-diagonal of their commutator matrix. We prove that, over a field of prime characterisitic, the coordinate ring of the latter is always $F$-regular and, with exactly one exception, so is that of the former, thus proving a conjecture of Kadyrsizova and Yerlanov., 11 pages, comments welcome!!!!

To every simple toric ideal $I_T$ one can associate the strongly robust simplicial complex $\Delta _T$, which determines the strongly robust property for all ideals that have $I_T$ as their bouquet ideal. We show that for the simple toric ideals of monomial curves in $\mathbb{A}^{s}$, the strongly robust simplicial complex $\Delta _T$ is either $\{\emptyset \}$ or contains exactly one 0-dimensional face. In the case of monomial curves in $\mathbb{A}^{3}$, the strongly robust simplicial complex $\Delta _T$ contains one 0-dimensional face if and only if the toric ideal $I_T$ is a complete intersection ideal with exactly two Betti degrees. Finally, we provide a construction to produce infinitely many strongly robust ideals with bouquet ideal the ideal of a monomial curve and show that they are all produced this way., Comment: 17 pages

We systematically study the intersection flatness and Ohm-Rush properties for modules over a commutative ring, drawing inspiration from the work of Ohm and Rush and of Hochster and Jeffries. We then use our newfound understanding of these properties to deduce consequences for the Frobenius map. We establish new structural results for modules that are intersection flat/Ohm-Rush by exhibiting intimate connections between these notions and the seminal work of Raynaud and Gruson on Mittag-Leffler modules. In particular, we develop a theory of Ohm-Rush modules that is parallel to the theory of Mittag-Leffler modules. Our motivation for doing so is that the Ohm-Rush condition is weaker than intersection flatness, and it will be shown in future work that one only needs the Ohm-Rush property of Frobenius on an excellent regular ring to prove the existence of test elements for homomorphic images of the ring. We also obtain descent and local-to-global results for intersection flat/Ohm-Rush modules. Our investigations reveal a pleasing picture for flat modules over a complete local ring, in which case many otherwise distinct properties coincide. Using these results, we characterize when a regular local ring has intersection flat Frobenius in terms of a purity condition on the relative Frobenius of the completion map, we provide a new characterization of excellence for DVRs in prime characteristic, we prove new cases of intersection flatness of Frobenius, we prove descent and ascent results for intersection flatness and Ohm-Rush properties for Frobenius, and we show that proving the existence of test elements for homomorphic images of excellent regular rings reduces to exhibiting openness of pure loci for certain simple module maps. We study this openness of loci problem in a special case and show that the Frobenius of any excellent regular ring of prime characteristic is close to being Ohm-Rush., Comments welcome!

We study the 0-dimensional schemes supported at one point in $n$-space which are $m$-symmetric, i.e. they intersect any curves thru the point with length $m$. We show that the maximal length for such a scheme is $m^n$ ($m$-superfat points) and we study properties of such schemes, in particular for $n=2$. We also study varieties defined by such schemes on Veronese and Segre-veronese varieties., 24 pages, 9 figures

We show that the perfectoidization can be (almost) calculated by using uniform completion or $p$-root closure in certain cases, including the semiperfectoid case. To do this, we focus on the universality of perfectoidization and uniform completion, as well as the $p$-root closed property of perfectoid rings. Through this calculation, we establish a connection between Roberts' classical closure operation ``$p$-root closure'' and Bhatt-Scholze's more recent concept of ``perfectoidization''., 18 pages, minor change

This is a survey article featuring some of Wolmer Vasconcelos contributions to commutative algebra, and explaining how Vasconcelos' work and insights have contributed to the development of commutative algebra and its interaction with other areas to the present. We show that the regularity of subrings of normal $k$-uniform monomial ideals is a monotone function, and we give a normality criterion for edge ideals of graphs using Ehrhart rings.

We develop Grothendieck's theory of dualizing complexes on finite posets, and its subsequent theory of Cohen-Macaulayness.

A net in $\mathbb{P}^2$ is a configuration of lines $\mathcal A$ and points $X$ satisfying certain incidence properties. Nets appear in a variety of settings, ranging from quasigroups to combinatorial design to classification of Kac-Moody algebras to cohomology jump loci of hyperplane arrangements. For a matroid $M$ and rank $r$, we associate a monomial ideal (a monomial variant of the Orlik-Solomon ideal) to the set of flats of $M$ of rank $\le r$. In the context of line arrangements in $\mathbb{P}^2$, applying Alexander duality to the resulting ideal yields insight into the combinatorial structure of nets., 15pp

In this paper, we classify several subcategories of the category of coherent sheaves on a noetherian divisorial scheme (e.g. a quasi-projective scheme over a commutative noetherian ring). More precisely, we classify the torsionfree (resp. torsion) classes closed under tensoring with line bundles by the subsets (resp. specialization-closed subsets) of the scheme, which generalizes the classification of torsionfree (resp. torsion) classes of the category of finitely generated modules over a commutative noetherian ring by Takahashi (resp. Stanley--Wang). Furthermore, we classify the Serre subcategories of a torsionfree class (in the sense of Quillen's exact categories) by using the above classifications, which gives a certain generalization of Gabriel's classification of Serre subcategories. As explicit applications, we classify the Serre subcategories of the category of maximal pure sheaves, which are a natural generalization of vector bundles for reducible schemes, on a reduced projective curve over a field, and the category of maximal Cohen-Macaulay modules over a one-dimensional Cohen-Macaulay ring., 25 pages, ver 2: added hyper references and corrected typos, comments welcome!

The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Musta\c{t}\u{a}, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic. In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical invariants defined via the Frobenius map. We also obtain a number of new results and simplified arguments in the regular case., Comment: v2: minor fixes after referee report; 58 pages; comments appreciated

In this paper we define and explore the analytic spread $\ell(\mathcal I)$ of a filtration in a local ring. We show that, especially for divisorial and symbolic filtrations, some basic properties of the analytic spread of an ideal extend to filtrations, even when the filtration is non Noetherian. We also illustrate some significant differences between the analytic spread of a filtration and the analytic spread of an ideal with examples. In the case of an ideal $I$, we have the classical bounds $\mbox{ht}(I)\le\ell(I)\le \dim R$. The upper bound $\ell(\mathcal I)\le \dim R$ is true for filtrations $\mathcal I$, but the lower bound is not true for all filtrations. We show that for the filtration $\mathcal I$ of symbolic powers of a height two prime ideal $\mathfrak p$ in a regular local ring of dimension three (a space curve singularity), so that $\mbox{ht}(\mathcal I) =2$ and $\dim R=3$, we have that $0\le \ell(\mathcal I)\le 2$ and all values of 0,1 and 2 can occur. In the cases of analytic spread 0 and 1 the symbolic algebra is necessarily non-Noetherian. The symbolic algebra is non-Noetherian if and only if $\ell(\mathfrak p^{(n)})=3$ for all symbolic powers of $\mathfrak p$ and if and only if $\ell(\mathcal I_a)=3$ for all truncations $\mathcal I_a$ of $\mathcal I$., Comment: 23 pages. Final version. Theorem 4.6 is a little stronger

In this paper we study the set of tensors that admit a special type of decomposition called an orthogonal tensor train decomposition. Finding equations defining varieties of low-rank tensors is generally a hard problem, however, the set of orthogonally decomposable tensors is defined by appealing quadratic equations. The tensors we consider are an extension of orthogonally decomposable tensors. We show that they are defined by similar quadratic equations, as well as an interesting higher-degree additional equation., 13 pages, 5 figures, Submitted to Vietnam Journal of Mathematics

A theorem due to Kazhdan and Ziegler implies that, by substituting linear forms for its variables, a homogeneous polynomial of sufficiently high strength specialises to any given polynomial of the same degree in a bounded number of variables. Using entirely different techniques, we extend this theorem to arbitrary polynomial functors. As a corollary of our work, we show that specialisation induces a quasi-order on elements in polynomial functors, and that among the elements with a dense orbit there are unique smallest and largest equivalence classes in this quasi-order., Comment: 19 pages

In this paper, we prove that if a $3$-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>3$ has only log canonical singularities, then so does a general hyperplane section $H$ of $X$. We also show that the same is true for klt singularities, which is a slight extension of \cite{ST20}. In the course of the proof, we provide a sufficient condition for log canonical (resp.~klt) surface singularities to be geometrically log canonical (resp.~geometrically klt) over a field., 28pages, 4figures

Interpreting the syzygy theorem for tame modules over posets in the setting of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning the existence of stratifications of real vector spaces that play well with sheaves having microsupport in a given cone or, equivalently, sheaves in the corresponding conic topology., Comment: 17 pages. Supersedes portion of arXiv:1908.09750 related to conjectures by Kashiwara and Schapira. Material in arXiv:1908.09750 on homological algebra over arbitrary posets and primary decomposition in partially ordered groups now in separate manuscripts. v2: updated references. v3: added alternate "constructible conic sheaf" versions of main results (4.5, 5.1, 5.2); expositional improvements

Over a complete Noetherian local domain of mixed characteristic with perfect residue field, we construct a perfectoid ring which is similar to an explicit representation of a perfect closure in positive characteristic. Then we demonstrate that this perfectoid ring is almost Cohen-Macaulay in the sense of almost ring theory. The proof of this result uses Andr\'e's flatness lemma along with Riemann's extension theorem. We stress that the idea partially originates from the ``perfectoidization" in the theory of prismatic cohomology., Comment: comments are welcome

We give a short, new proof of a recent result of Hanlon-Hicks-Lazarev about toric varieties. As in their work, this leads to a proof of a conjecture of Berkesch-Erman-Smith on virtual resolutions and to a resolution of the diagonal in the simplicial case.