Search

We extend the theory of Koszul modules to the bi-graded case, and prove a vanishing theorem that allows us to show that the Canonical Ribbon Conjecture of Bayer and Eisenbud holds over a field of characteristic zero or at least equal to the Clifford index. Our results confirm a conjecture of Eisenbud and Schreyer regarding the characteristics where the generic statement of Green's conjecture holds. They also recover and extend to positive characteristics results due to Aprodu and Voisin asserting that Green's Conjecture holds for generic curves of each gonality., Comment: 23 pages

Let R be a regular local ring. Let G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle over R is trivial if it is trivial over the quotient field of R. The conjecture is known when R contains a field. We prove the conjecture for a large class of regular local rings not containing fields in the case when G is split., Comment: The final version to be published in Transactions of the AMS. Results about quadratic forms are strengthened. In the section on Bertini type theorems a correction in the case of a non-perfect residue field is made. Other minor corrections and improvements

When studying a graded module $M$ over the Cox ring of a smooth projective toric variety $X$, there are two standard types of resolutions commonly used to glean information: free resolutions of $M$ and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study. The theory of virtual resolutions includes a notion of a virtually Cohen--Macaulay property, though tools for assessing which modules are virtually Cohen--Macaulay have only recently started to be developed. In this paper, we continue this research program in two related ways. The first is that, when $X$ is a product of projective spaces, we produce a large new class of virtually Cohen--Macaulay Stanley--Reisner rings, which we show to be virtually Cohen--Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety $X$, we develop homological tools for assessing the virtual Cohen--Macaulay property. Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. We also use these tools to establish relationships among the arithmetically, geometrically, and virtually Cohen--Macaulay properties., Accepted to Transactions of the London Mathematical Society

We prove local uniformization of Abhyankar valuations of an algebraic function field K over a ground field k. Our result generalizes the proof of this result, with the additional assumption that the residue field of the valuation ring is separable over k, by Hagen Knaf and Franz-Viktor Kuhlmann. The proof in this paper uses different methods, being inspired by the approach of Zariski and Abhyankar., Comment: 28 pages. Final Final version

We introduce a "workable" notion of degree for non-homogeneous polynomial ideals and formulate and prove ideal theoretic B\'ezout Inequalities for the sum of two ideals in terms of this notion of degree and the degree of generators. We compute probabilistically the degree of an equidimensional ideal.

Let $S$ be an unramified regular local ring of mixed characteristic two and $R$ the integral closure of $S$ in a biquadratic extension of its quotient field obtained by adjoining roots of sufficiently general square free elements $f,g\in S$. Let $S^2$ denote the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/2S$. When at least one of $f,g\in S^2$, we characterize the Cohen-Macaulayness of $R$ and show that $R$ admits a birational small Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay in case $f,g\in S^2$ or if $f,g\notin S^2$., Comment: Final version, to appear in Journal of Algebra; minor changes, unabbreviated title, updated references

Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no three lines on it form a “triangle”.

We study the behavior of representation varieties of quivers with relations under the operation of node splitting. We show how splitting a node gives a correspondence between certain closed subvarieties of representation varieties for different algebras, which preserves properties like normality or having rational singularities. Furthermore, we describe how the defining equations of such closed subvarieties change under the correspondence. By working in the "relative setting" (splitting one node at a time), we demonstrate that there are many non-hereditary algebras whose irreducible components of representation varieties are all normal with rational singularities. We also obtain explicit generators of the prime defining ideals of these irreducible components. This class contains all radical square zero algebras, but also many others, as illustrated by examples throughout the paper. We also show the above is true when replacing irreducible components by orbit closures, for a more restrictive class of algebras. Lastly, we provide applications to decompositions of moduli spaces of semistable representations of certain algebras., 26 pages. Updated references, typos, metadata. Actually final version

In this article, we establish a motivic analog of an enumeration result of James-Thomas on non-stable vector bundles in topological setting. Using this, we obtain results on enumeration of projective modules of rank $d$ over a smooth affine $k$-algebra $A$ of dimension $d$, recovering in particular results of Suslin and Bhatwadekar on cancellation of such vector bundles. Admitting a conjecture of Asok and Fasel, we prove cancellation of such vector bundles of rank $d-1$ if the base field $k$ is algebraically closed. We also explore the cancellation properties of symplectic vector bundles., Comment: New version, 37 pages. Added a cancellation result for symplectic vector bundles and exposition improved. Comments are welcome!

We consider blowups at a general point of weighted projective planes and, more generally, of toric surfaces with Picard number one. We give a unifying construction of negative curves on these blowups such that all previously known families appear as boundary cases of this. The classification consists of two classes of said curves, each depending on two parameters. Every curve in these two classes is algebraically related to other curves in both classes; this allows us to find their defining equations inductively. For each curve in our classification, we consider a family of blowups in which the curve defines an extremal class in the effective cone. We give a complete classification of these blowups into Mori Dream Spaces and non-Mori Dream Spaces. Our approach greatly simplifies previous proofs, avoiding positive characteristic methods and higher cohomology., 21 pages, 3 figures

In this article, we study certain local cohomology modules over $F$-pure rings. We give sufficient conditions for the vanishing of some Lyubeznik numbers, derive a formula for computing these invariants when the $F$-pure ring is standard graded and, by its means, we provide some new examples of Lyubeznik tables. We study associated primes of certain Ext-modules, showing that they are all compatible ideals. Finally, we focus on properties that Lyubeznik numbers detect over a globally $F$-split projective variety., Comment: Fixed errors in the proof of Lemma 3.7 and in Example 4.11 of previous version

Musta\c{t}\u{a} defined Bernstein-Sato polynomials in prime characteristic for principal ideals and proved that the roots of these polynomials are related to the $F$-jumping numbers of the ideal. This approach was later refined by Bitoun. Here we generalize these techniques to develop analogous notions for the case of arbitrary ideals and prove that these have similar connections to $F$-jumping numbers., Comment: v2: added Subsections 6.1 (some remarks about compatibility with char. 0) 6.5 (other characterizations of Bernstein-Sato roots) and 6.6 (examples). Relaxed assumptions on R. 33 pages

We use an algebraic approach to construct minimal decompositions of symmetric tensors with low rank. This is done by using Apolarity Theory and by studying minimal sets of reduced points apolar to a given symmetric tensor, namely, whose ideal is contained in the apolar ideal associated to the tensor. In particular, we focus on the structure of the Hilbert function of these ideals of points. We give a procedure which produces a minimal set of points apolar to any symmetric tensor of rank at most 5. This procedure is also implemented in the algebra software Macaulay2., 26 pages; the Macaulay2 package "ApolarLowRank.m2" is available in the ancillary files

We study cubic surfaces as symmetric tensors of format $4 \times 4 \times 4$. We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The corresponding algebraic problem concerns border ranks. We show that the non-symmetric border rank coincides with the symmetric border rank for cubic surfaces. As part of our analysis, we obtain minimal ideal generators for the symmetric analogue to the secant variety from the salmon conjecture. We also give a test for symmetric rank given by the non-vanishing of certain discriminants. The results extend to order three tensors of all sizes, implying the equality of rank and symmetric rank when the symmetric rank is at most seven, and the equality of border rank and symmetric border rank when the symmetric border rank is at most five. We also study real ranks via the real substitution method., 16 pages

Previously, Ohsugi and Hibi gave a combinatorial description of bipartite graphs $G$ whose toric edge ideal $I_G$ is generated by quadrics, showing that every cycle of $G$ of length at least $6$ must have a chord. This corresponds to the Green-Lazarsfeld condition $\mathbf{N}_1$. In this paper, we investigate the higher syzygies of $I_G$ and give combinatorial descriptions of the Green-Lazarsfeld conditions $\mathbf{N}_p$ of toric edge ideals of bipartite graphs for all $p \ge 1$. In particular, we show that $I_G$ is linearly presented (i.e. satisfies condition $\mathbf{N}_2$) if and only if the bipartite complement of $G$ is a tree of diameter at most $3$. We also investigate the regularity of linearly presented toric edge ideals and give criteria for polyomino ideals to satisfy the Green-Lazarsfeld conditions., 20 pages, comments welcome

In the present article, we investigate the following deformation problem. Let $(R,\mathfrak m)$ be a local (graded local) Noetherian ring with a (homogeneous) regular element $y \in \mathfrak m$ and assume that $R/yR$ is quasi-Gorenstein. Then is $R$ quasi-Gorenstein? We give positive answers to this problem under various assumptions, while we present a counter-example in general. We emphasize that absence of the Cohen-Macaulay condition requires some delicate studies., Comment: To appear in J. Algebra

Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in $\mathbb P_k^n$; additional interest come from their rich structure, which allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra and representation theory. In the present paper we investigate the more general problem of determining the structure of symbolic powers of a wide generalization of star configurations of points (introduced by Geramita, Harbourne, Migliore and Nagel) called star configurations of hypersurfaces in $\mathbb P_k^n$. Here (1) we provide explicit minimal generating sets of the symbolic powers $I^{(m)}$ of these ideals $I$, (2) we introduce a notion of $\delta$-c.i. quotients, which generalize ideals with linear quotients, and show that $I^{(m)}$ have $\delta$-c.i. quotients, (3) we show that the shape of the Betti tables of these symbolic powers is determined by certain "Koszul" strands and we prove that a little bit more than the bottom half of the Betti table has a regular, almost hypnotic, pattern, and (4) we provide a closed formula for all the graded Betti numbers in these strands. As a special case of (2) we deduce that symbolic powers of ideals of star configurations of points have linear quotients. We also improve and extend results by Galetto, Geramita, Shin and Van Tuyl, and provide explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations. Finally, inspired by Young tableaux, we introduce a technical tool which may be of independent interest: it is a "canonical" way of writing any monomial in any given set of polynomials. Our methods are characteristic--free., Comment: Final revision (original paper was accepted for publication in Trans. Amer. Math. Soc.)

Hyperplane Arrangements of rank $3$ admitting an unbalanced Ziegler restriction are known to fulfill Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this note, we prove that arrangements that admit a locally heavy flag satisfy Terao's conjecture which is a generalization of the statement above to arbitrary dimension. To this end, we extend results characterizing the freeness of multiarrangements with a heavy hyperplane to those satisfying the weaker notion of a locally heavy hyperplane. As a corollary, we give a new proof that irreducible arrangements with a generic hyperplane are totally non-free. In another application, we show that an irreducible multiarrangement of rank $3$ with at least two locally heavy hyperplanes is not free., Comment: 16 pages, 1 figure. arXiv admin note: text overlap with arXiv:1603.05803

Given a finite order ideal $\mathcal O$ in the polynomial ring $K[x_1,\dots, x_n]$ over a field $K$, let $\partial \mathcal O$ be the border of $\mathcal O$ and $\mathcal P_{\mathcal O}$ the Pommaret basis of the ideal generated by the terms outside $\mathcal O$. In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $\partial\mathcal O$-marked sets (resp. bases) and $\mathcal P_{\mathcal O}$-marked sets (resp. bases). We prove that a $\partial\mathcal O$-marked set $B$ is a marked basis if and only if the $\mathcal P_{\mathcal O}$-marked set $P$ contained in $B$ is a marked basis and generates the same ideal as $B$. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $\partial\mathcal O$-marked bases and $\mathcal P_{\mathcal O}$-marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gr\"obner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in smaller affine spaces. Furthermore, we observe that Pommaret marked schemes give an open covering of punctual Hilbert schemes. Several examples are given along all the paper., Comment: 17 pages; presentation improved, some references added

We introduce a relativization of the secant sheaves used by Ein, Green and Lazarsfeld and apply this construction to the study of syzygies of canonical curves. As a first application, we give a simpler proof of Voisin's Theorem for general canonical curves. This completely determines the terms of the minimal free resolution of the coordinate ring of such curves. Secondly, in the case of curves of even genus, we enhance Voisin's Theorem by providing a structure theorem for the last syzygy space, resolving the Geometric Syzygy Conjecture in even genus., Comment: Final version, to appear in Inventiones Math. The statements concerning positive characteristic have been removed and will be published separately. This submission supersedes arXiv:1910.06200 and arXiv:1907.07553

In this article, we study the weak and strong Lefschetz properties, and the related notion of almost revlex ideal, in the non-Artinian case, proving that several results known in the Artinian case hold also in this more general setting. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements., To appear in the Journal of Algebra

Hartshorne conjectured that a smooth, codimension c subvariety of n-dimensional projective space must be a complete intersection, whenever c is less than n/3. We prove this in the special case when n is much larger than the degree of the subvariety. Similar results were known in characteristic zero due to Hartshorne, Barth-Van de Ven, and others. Our proof is field independent and employs quite different methods from those previous results, as we connect Hartshorne's Conjecture with the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's Conjecture., Comment: 5 pages

We present a classification algorithm for isolated hypersurface singularities of corank 2 and modality 1 over the real numbers. For a singularity given by a polynomial over the rationals, the algorithm determines its right equivalence class by specifying all representatives in Arnold's list of normal forms (Arnold et al. 1985) belonging to this class, and the corresponding values of the moduli parameter. We discuss how to computationally realize the individual steps of the algorithm for all singularities in consideration, and give explicit examples. The algorithm is implemented in the Singular library realclassify.lib., 31 pages, 5 figures, 1 table, improvements in the algorithms

Let $\mathscr{R}$ denote the ring of real polynomials on $\mathbb{R}^{n}$. Fix $m\geq 0$, and let $A_{1},\cdots ,A_{M}\in \mathscr{R}$. The $ C^{m}$-closure of $\left( A_{1},\cdots ,A_{M}\right) $, denoted here by $ \left[ A_{1},\cdots ,A_{M};C^{m}\right] $, is the ideal of all $f\in \mathscr{R}$ expressible in the form $f=F_{1}A_{1}+\cdots +F_{M}A_{M}$ with each $F_{i}\in C^{m}\left( \mathbb{R}^{n}\right) $. In this paper we exhibit an algorithm to compute generators for $\left[ A_{1},\cdots ,A_{M};C^{m}\right] $., 47 pages, see also the related article "Solutions to a System of Equations for $C^m$ Functions"

What kind of reduced monomial schemes can be obtained as a Gr\"obner degeneration of a smooth projective variety? Our conjectured answer is: only Stanley-Reisner schemes associated to acyclic Cohen-Macaulay simplicial complexes. This would imply, in particular, that only curves of genus zero have such a degeneration. We prove this conjecture for degrevlex orders, for elliptic curves over real number fields, for boundaries of cross-polytopes, and for leafless graphs. We discuss consequences for rational and F-rational singularities of algebras with straightening laws., Comment: 12 pages

By extending some basic results of Grothendieck and Foxby about local cohomology to commutative DG-rings, we prove new amplitude inequalities about finite DG-modules of finite injective dimension over commutative local DG-rings, complementing results of J{\o}rgensen and resolving a recent conjecture of Minamoto. When these inequalities are equalities, we arrive to the notion of a local-Cohen-Macaulay DG-ring. We make a detailed study of this notion, showing that much of the classical theory of Cohen-Macaulay rings and modules can be generalized to the derived setting, and that there are many natural examples of local-Cohen-Macaulay DG-rings. In particular, local Gorenstein DG-rings are local-Cohen-Macaulay. Our work is in a non-positive cohomological situation, allowing the Cohen-Macaulay condition to be introduced to derived algebraic geometry, but we also discuss extensions of it to non-negative DG-rings, which could lead to the concept of Cohen-Macaulayness in topology., Comment: 41 pages, final version, to appear in Transactions of the AMS

In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of dimension one. We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation. Using results on such equations, we prove the convergence of the formal Puiseux series solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them. In addition, we bound the degree of the possible algebraic and rational solutions, and we provide an algorithm to decide their existence and to compute such solutions if they exist. Moreover, if the reduced differential equation is non trivial, for every given point (x0,y0)∈C2, we prove the existence of a convergent Puiseux series solution y(x) of the original system such that y(x0)=y0., Agencia Estatal de Investigación, Ministerio de Economía y Competitividad, Austrian Science Fund

Given a polynomial $f$ with coefficients in a field of prime characteristic $p$, it is known that there exists a differential operator that raises $1/f$ to its $p$th power. We first discuss a relation between the `level' of this differential operator and the notion of `stratification' in the case of hyperelliptic curves. Next we extend the notion of level to that of a pair of polynomials. We prove some basic properties and we compute this level in certain special cases. In particular we present examples of polynomials $g$ and $f$ such that there is no differential operator raising $g/f$ to its $p$th power., 14 pages, comments are welcome

Let $k$ be a field of positive characteristic. Building on the work of the second named author, we define a new class of $k$-algebras, called diagonally $F$-regular algebras, for which the so-called Uniform Symbolic Topology Property (USTP) holds effectively. We show that this class contains all essentially smooth $k$-algebras. We also show that this class contains certain singular algebras, such as the affine cone over $\mathbb{P}^r_{k} \times \mathbb{P}^s_{k}$, when $k$ is perfect. By reduction to positive characteristic, it follows that USTP holds effectively for the affine cone over $\mathbb{P}^r_{\mathbb{C}} \times \mathbb{P}^s_{\mathbb{C}}$ and more generally for complex varieties of diagonal $F$-regular type., Comment: 22 pages. Comments welcome

The theory of mixed multiplicities of filtrations by $m$-primary ideals in a ring is introduced in a recent paper by Cutkosky, Sarkar and Srinivasan. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed multiplicities of filtrations must be nonnegative real numbers and give examples to show that they could be zero or even irrational. When $R$ is analytically irreducible, and $\mathcal I(1),\ldots,\mathcal I(r)$ are filtrations of $R$ by $m_R$-primary ideals, we show that all of the mixed multiplicities $e_R(\mathcal I(1)^{[d_1]},\ldots,\mathcal I(r)^{[d_r]};R)$ are positive if and only if the ordinary multiplicities $e_R(\mathcal I(i);R)$ for $1\le i\le r$ are positive. We extend this to modules and prove a simple characterization of when the mixed multiplicities are positive or zero on a finitely generated module., 15 pages

We exhibit a new subadditivity formula for test ideals on singular varieties using an argument similar to those of Demailly-Ein-Lazarsfeld and Hara-Yoshida. Any subadditivity formula for singular varieties must have a correction term that measures the singularities of that variety. Whereas earlier subadditivity formulas accomplished this by multiplying by the Jacobian ideal, our approach is to use the formalism of Cartier algebras. We also show that our subadditivity containment is sharper than ones shown previously by Takagi and Eisenstein. The first of these results follows from a Noether normalization technique due to Hochster and Huneke. The second of these results is obtained using ideas of Takagi and Eisenstein to show that the adjoint ideal $\mathscr{J}_X(A, Z) $ reduces mod $p$ to Takagi's adjoint test ideal, even when the ambient space is singular, provided that $A$ is regular at the generic point of $X$. One difficulty of using this new subadditivity formula in practice is the computational complexity of computing its correction term. Thus, we discuss a combinatorial construction of the relevant Cartier algebra in the toric setting., Comment: Many changes big and small. Most notably, the proofs of theorems 3.12 and 5.15 were corrected in response to referee feedback. I also cleaned up the notation in the proof of 5.15

This note describes a package for computing seminormalization of rings within Macaulay2., Comment: 6 pages, comments welcome, the latest version of the package is available in https://github.com/kschwede/M2/blob/master/M2/Macaulay2/packages/Seminormalization.m2

We show that the Waring rank of the $3 \times 3$ determinant, previously known to be between $14$ and $18$, is at least $15$. We use syzygies of the apolar ideal, which have not been used in this way before. Additionally, we show that the cactus rank of the $3 \times 3$ permanent is at least $14$., 17 pages. v2: clarifications and additional references. To appear in Linear Algebra Appl

We construct irreducible complex projective varieties such that the Lyubeznik numbers of their affine cones depend on the choices of projective embeddings. The main ingredient is the recent work of Reichelt-Saito-Walther, where the Lyubeznik numbers are reinterpreted using perverse sheaves and reducible projective varieties with dependent Lyubeznik numbers are constructed.

In this paper, we characterize the (generalized) Frobenius powers and critical exponents of two classes of monomial ideals of a polynomial ring in positive characteristic: powers of the homogeneous maximal ideal, and ideals generated by positive powers of the variables. In doing so, we effectively characterize the test ideals and $F$-jumping exponents of sufficiently general homogeneous polynomials, and of all diagonal polynomials. Our characterizations make these invariants computable, and show that they vary uniformly with the congruence class of the characteristic modulo a fixed integer. Moreover, we confirm that for a diagonal polynomial over a field of characteristic zero, the test ideals of its reduction modulo a prime agree with the reductions of its multiplier ideals for infinitely many primes., Comment: 19 pages, 1 figure, comments welcome

This article extends the notion of a Frobenius power of an ideal in prime characteristic to allow arbitrary nonnegative real exponents. These generalized Frobenius powers are closely related to test ideals in prime characteristic, and multiplier ideals over fields of characteristic zero. For instance, like these well-known families of ideals, Frobenius powers also give rise to jumping exponents that we call critical Frobenius exponents. In fact, the Frobenius powers of a principal ideal coincides with its test ideals, but appear to be a more refined measure of singularities in general. Herein, we develop the theory of Frobenius powers in regular domains, and apply it to study singularities, especially those of generic hypersurfaces. These applications illustrate one way in which multiplier ideals behave more like Frobenius powers than like test ideals., Minor updates based on anonymous referee's comments

We study systems of parameters over finite fields from a probabilistic perspective, and use this to give the first effective Noether normalization result over a finite field. Our central technique is an adaptation of Poonen's closed point sieve, where we sieve over higher dimensional subvarieties, and we express the desired probabilities via a zeta function-like power series that enumerates higher dimensional varieties instead of closed points. This also yields a new proof of a recent result of Gabber-Liu-Lorenzini and Chinburg-Moret-Bailly-Pappas-Taylor on Noether normalizations of projective families over the integers., 20 pages. Minor revisions to exposition

A weak wild arithmetic quotient singularity arises from the quotient of a smooth arithmetic surface by a finite group action, where the inertia group of a point on a closed characteristic p fiber is a p-group acting with smallest possible ramification jump. In this paper, we give complete explicit resolutions of these singularities using deformation theory and valuation theory, taking a more local perspective than previous work has taken. Our descriptions answer several questions of Lorenzini. Along the way, we give a valuation-theoretic criterion for a normal snc-model of P^1 over a discretely valued field to be regular., Final version, to appear in the Journal of Algebraic Geometry. 31 pages

"This is the peer reviewed version of the following article: Abad C, Bravo A, Villamayor U. OV. Finite morphisms and simultaneous reduction of the multiplicity. Mathematische Nachrichten. 2020;293:8–38, which has been published in final form at https://doi.org/10.1002/mana.201800470. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited", Let X be a singular algebraic variety defined over a perfect field k, with quotient field K ( X ) . Let s ≥ 2 be the highest multiplicity of X and let F s ( X ) be the set of points of multiplicity s. If Y ⊂ F s ( X ) is a regular center and X ← X 1 is the blow up at Y, then the highest multiplicity of X1 is less than or equal to s. A sequence of blow ups at regular centers Y i ⊂ F s ( X i ) , say X ← X 1 ← ⋯ ← X n , is said to be a simplification of the multiplicity if the maximum multiplicity of X n is strictly lower than that of X, that is, if F s ( X n ) is empty. In characteristic zero there is an algorithm which assigns to each X a unique simplification of the multiplicity. However, the problem remains open when the characteristic is positive. In this paper we will study finite dominant morphisms between singular varieties β : X ′ → X of generic rank r ≥ 1 (i.e., [ K ( X ′ ) : K ( X ) ] = r ). We will see that, when imposing suitable conditions on β, there is a strong link between the strata of maximum multiplicity of X and X ′ , say F s ( X ) and F r s ( X ′ ) respectively. In such case, we will say that the morphism is strongly transversal. When β : X ′ → X is strongly transversal one can obtain information about the simplification of the multiplicity of X from that of X ′ and vice versa. Finally, we will see that given a singular variety X and a finite field extension L of K ( X ) of rank r ≥ 1 , one can construct (at least locally, in étale topology) a strongly transversal morphism β : X ′ → X , where X ′ has quotient field L., The authors were partially supported by the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R&D (SEV2015-0554), and through MTM2015-68524-P (MINECO/FEDER)

We determine the equations of the blowup of $\mathbb{P}^n$ along a $d$-fold rational normal scroll $S$, and we prove that the Rees ring and special fiber ring of $S \subseteq \mathbb{P}^n$ are Koszul algebras., To appear on Transactions of the American Mathematical Society

Let $X$ be the blowup of a weighted projective plane at a general point. We study the problem of finite generation of the Cox ring of $X$. Generalizing examples of Srinivasan and Kurano-Nishida, we consider examples of $X$ that contain a negative curve of the class $H-mE$, where $H$ is the class of a divisor pulled back from the weighted projective plane and $E$ is the class of the exceptional curve. For any $m>0$ we construct examples where the Cox ring is finitely generated and examples where it is not., 19 pages, 9 figures

We define a linear code $C_\eta(\delta_T,\delta_X)$ by evaluating polynomials of bidegree $(\delta_T,\delta_X)$ in the Cox ring on $\mathbb{F}_q$-rational points of the Hirzebruch surface of parameter $\eta$ on the finite field $\mathbb{F}_q$. We give explicit parameters of the code, notably using Gr\"obner bases. The minimum distance provides an upper bound of the number of $\mathbb{F}_q$-rational points of a non-filling curve on a Hirzebruch surface., Comment: Revised version [11/2018]

Let k be a field of characteristic zero. This paper studies a problem proposed by Joseph F. Ritt in 1950. Precisely, we prove that (1) If p ⩾ 2 is an integer, for every integer i ∈ N , the nilpotency index of the image of T i in the ring k { T } / [ T p ] equals ( i + 1 ) p − i . (2) For every pair of integers ( i , j ) , the nilpotency index of the image of T i U j in the ring k { T } / [ T U ] equals i + j + 1 .

We propose a general definition of mathematical instanton bundle with given charge on any Fano threefold extending the classical definitions on $\mathbb P^3$ and on Fano threefold with cyclic Picard group. Then we deal with the case of the blow up of $\mathbb P^3$ at a point, giving an explicit construction of instanton bundles satisfying some important extra properties: moreover, we also show that they correspond to smooth points of a component of the moduli space., Comment: 24 pages. Final version in the Michigan Mathematical Journal

In this paper, we give a detailed proof to a result of Gabber (unpublished) on the lifting problem of quasi-excellent rings, extending the previous work on Nishimura-Nishimura. As a corollary, we establish that an ideal-adic completion of an excellent (resp. quasi-excellent) ring is excellent (resp. quasi-excellent)., The final version, accepted by Kyoto J. Math

We prove stability of logarithmic tangent sheaves of singular hypersurfaces D of the projective space with constraints on the dimension and degree of the singularities of D. As main application, we prove that determinants and symmetric determinants have stable logarithmic tangent sheaves and we describe an open dense piece of the associated moduli space., Major update. To appear in IMRN

Every tropical ideal in the sense of Maclagan-Rinc\'on has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the V\'amos matroid and the uniform matroid of rank two on three elements, and in which all maximal cones have weight one., Comment: 11 pages; v2 includes two short new sections, main theorem is stated in slightly more generality

We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is $F$-finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, K\"uronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama's description of the restricted base locus to klt or strongly $F$-regular varieties over arbitrary fields., Comment: 27 pages; comments welcome. v2: Added Appendix B, other small changes. v3: Added Remark 4.4 and Proposition 4.6, updated references, other small changes

In this paper, we study the class of free hyperplane arrangements. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over $$\mathbb {Q}$$ .

We analyze the vanishing and non-vanishing behavior of the graded Betti numbers for Segre embeddings of products of projective spaces. We give lower bounds for when each of the rows of the Betti table becomes non-zero, and prove that our bounds are tight for Segre embeddings of products of P^1. This generalizes results of Rubei concerning the Green-Lazarsfeld property N_p for Segre embeddings. Our methods combine the Kempf-Weyman geometric technique for computing syzygies, the Ein-Erman-Lazarsfeld approach to proving non-vanishing of Betti numbers, and the theory of algebras with straightening laws.