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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!

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

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

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

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

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

If $R$ is a topological ring, then it is well known that $R^{\ast}$, the group of units of $R$, with the subspace topology is not necessarily a topological group. This fact first leads us to a natural definition: By an \emph{absolute topological ring} we mean a topological ring such that its group of units with the subspace topology is a topological group. We prove that every commutative ring with the $I$-adic topology is an absolute topological ring. Next we show that for a given topological ring $R$ then $R^{\ast}$ with the subspace topology $\mathscr{T}$ is a topological group (or equivalently, $R$ is an absolute topological ring) if and only if $\mathscr{T}=\mathscr{T}_{f}$ where the topology $\mathscr{T}_{f}$ over $R^{\ast}$ is induced by the map $R^{\ast}\rightarrow R\times R$ which is given by $a\mapsto(a,a^{-1})$. If $G$ is a topological group then every monomial function $G^{n}\rightarrow G$ as well as if $R$ is a topological ring then every polynomial function $R^{n}\rightarrow R$ are continuous. In particular, the Boolean ring of every topological ring with the subspace topology is a topological ring. We prove that for the $I$-adic topology over a ring $R$, then $\pi_{0}(R)=R/(\bigcap\limits_{n\geqslant1}I^{n})=t(R)$ where $\pi_{0}(R)$ is the space of connected components of $R$ and $t(R)$ is the space of irreducible closed subsets of $R$. We show that if the identity element of a topological group is dense, then its topology is trivial. As a consequence, a normal subgroup of a topological group is dense if and only if the topology of the quotient group is trivial. Finally, we realized that the main result of Koh \cite{kwangil} as well as its corrected version \cite[Chap II, \S12, Theorem 12.1]{Ursul} are not true, then we corrected this result in the right way., Comment: 13 pages

In this paper, we prove that klt singularities are invariant under deformations if the generic fiber is $\mathbb{Q}$-Gorenstein. We also obtain a similar result for slc singularities. These are generalizations of results of Esnault-Viehweg and S. Ishii., 42pages; v2: minor changes. arXiv admin note: text overlap with arXiv:2103.03721

Weighted projective spaces are natural generalizations of projective spaces with a rich structure. Projective Reed-Muller codes are error-correcting codes that played an important role in reliably transmitting information on digital communication channels. In this case study, we explore the power of commutative and homological algebraic techniques to study weighted projective Reed-Muller (WPRM) codes on weighted projective spaces of the form $\mathbb{P}(1,1,a)$. We compute minimal free resolutions and thereby obtain Hilbert series for the vanishing ideal of the $\mathbb{F}_q$-rational points, and compute main parameters for these codes.

Let $X$ be a toric variety. We establish vanishing (and non-vanishing) results for the sheaves $R^if_*\Omega^p_{\tilde X}(\log E)$, where $f: \tilde{X} \to X$ is a strong log resolution of singularities with reduced exceptional divisor $E$. These extend the local vanishing theorem for toric varieties in [MOP20]. Our consideration of these sheaves is motivated by the notion of $k$-rational singularities introduced by Friedman and Laza [FL22b]. In particular, our results lead to criteria for toric varieties to have $k$-rational singularities, as defined in [SVV23]., Comment: 17 pages. Comments welcome!

This paper establishes semiorthogonal decompositions for derived Grassmannians of perfect complexes with Tor-amplitude in $[0,1]$. This result verifies the author's Quot formula conjecture [J21a] and generalizes and strengthens Toda's result in [Tod23]. We give applications of this result to various classical situations such as blowups of determinantal ideals, reducible schemes, and varieties of linear series on curves. Our approach utilizes the framework of derived algebraic geometry, allowing us to work over arbitrary base spaces over $\mathbb{Q}$. It also provides concrete descriptions of Fourier-Mukai kernels in terms of derived Schur functors., Comment: 26 pages. Comments welcome!

Building off of work of Harada, Nowroozi, and Van Tuyl which provided particular length two virtual resolutions for finite sets of points in $\mathbb{P}^1\times\mathbb{P}^1$, we prove that the vast majority of virtual resolutions of a pair for minimal elements of the multigraded regularity in this setting are of Hilbert-Burch type. We give explicit descriptions of these short virtual resolutions that depend only on the number of points. Moreover, despite initial evidence, we show that these virtual resolutions are not always short, and we give sufficient conditions for when they are length three., Comment: 21 pages, comments welcome!

A $\Bbbk$-configuration of type $(d_1,\dots,d_s)$ is a specific set of points in $\mathbb P^2$ that has a number of algebraic and geometric properties. For example, the graded Betti numbers and Hilbert functions of all $\Bbbk$-configurations in $\mathbb P^2$ are determined by the type $(d_1,\dots,d_s)$. However the Waldschmidt constant of a $\Bbbk$-configuration in $\mathbb P^2$ of the same type may vary. In this paper, we find that the Waldschmidt constant of a $\Bbbk$-configuration in $\mathbb P^2$ of type $(d_1,\dots,d_s)$ with $d_1\ge s\ge 1$ is $s$. We also find the Waldschmidt constant of a standard $\Bbbk$-configuration in $\mathbb P^2$ of type $(a,b,c)$ with $a\ge 1$ except the type $(2,3,5)$. In particular, we prove that the Waldschmidt constant of a standard $\Bbbk$-configuration in $\mathbb P^2$ of type $(1,b,c)$ with $c\ge 2b+2$ does not depend on $c$.

We characterize the number of points for which there exist non-empty Terracini sets of points in $\mathbb{P}^n$. Then we study minimally Terracini finite sets of points in $\mathbb{P}^n$ and we obtain a complete description in the case of $\mathbb{P}^3$, when the number of points is less than twice the degree of the linear system., Comment: 23 pages

We show that the shortest nonzero polynomials vanishing on bounded-rank matrices and skew-symmetric matrices are the determinants and Pfaffians characterising the rank. Algebraically, this means that in the ideal generated by all $t$-minors or $t$-Pfaffians of a generic matrix or skew-symmetric matrix one cannot find any polynomial with fewer terms than those determinants or Pfaffians, respectively, and that those determinants and Pfaffians are essentially the only polynomials in the ideal with that many terms. As a key tool of independent interest, we show that the ideal of a sufficiently general $t$-dimensional subspace of an affine $n$-space does not contain polynomials with fewer than $t+1$ terms., 13 pages, comments welcome, v2: 15 pages, final version as in Bulletin LMS

We show that if a finitely generated prime algebra $\Delta$ is a finitely generated maximal Cohen-Macaulay module over its centre $Z$, and has global dimension equal to $\dim Z$, then the pair given by its centre and ramification divisor is Kawamata log terminal.

We prove the existence of ternary forms admitting apolar sets of points of cardinality equal to the Waring rank, but having different Hilbert function and different regularity. This is done exploiting liaison theory and Cayley-Bacharach properties for sets of points in the projective plane, Comment: 12 pages. Comments are welcome!

Let $A$ be a two-dimensional excellent normal Gorenstein local domain. In this paper, we characterize elliptic ideals $I \subset A$ for its normal tangent cone $\overline{G}(I)$ to be Gorenstein. Moreover, we classify all those ideals in a Gorenstein elliptic singularity in the characteristic zero case., Comment: 20 pages

We associate to every graph a linear program for packings of vertex disjoint paths. We show that the optimal primal and dual values of the corresponding integer program are the binomial grade and height of the binomial edge ideal of the graph. We deduce from this a new combinatorial characterization of graphs of K\"onig type and use it to show that all trees are of K\"onig type. The log canonical threshold and the F-threshold are important invariants associated to the singularities of a variety in characteristic $0$ and characteristic $p$. We show that the optimal value of the linear program (computed over the rationals) agrees with both the F-threshold and the log canonical threshold of the binomial edge ideal if the graph is a block graph or of K\"onig type. We conjecture that this linear program computes the log canonical threshold of the binomial edge ideal of any graph. Our results resemble theorems on monomial ideals arising from hypergraphs due to Howald and others., Comment: updated introduction, added reference, other minor revisions

In 1966, David Mumford created a drawing of $\operatorname{Proj} \mathbb Z [X,Y]$ in his book, "Lectures on Curves on an Algebraic Surface". In following, he created a photo of a so-called 'arithmetic surface' $\operatorname{Spec} \mathbb Z [T]$for his 1988 book, "The Red Book of Varieties and Schemes". The depiction presents the structure of $\operatorname{Spec} \mathbb Z [T]$ as being interesting and pleasant and is a well-known picture in algebraic geometry. Taking inspiration from Mumford, we create a drawing similar to $\operatorname{Spec} \mathbb Z_p [T]$, which has a lot of similarities with $\operatorname{Spec} \mathbb Z [T]$., Comment: 8 pages

A Taylor variety consists of all fixed order Taylor polynomials of rational functions, where the number of variables and degrees of numerators and denominators are fixed. In one variable, Taylor varieties are given by rank constraints on Hankel matrices. Inversion of the natural parametrization is known as Pad\'e approximation. We study the dimension and defining ideals of Taylor varieties. Taylor hypersurfaces are interesting for projective geometry, since their Hessians tend to vanish. In three and more variables, there exist defective Taylor varieties whose dimension is smaller than the number of parameters. We explain this with Fr\"oberg's Conjecture in commutative algebra., Comment: 20 pages

A chopped ideal is obtained from a homogeneous ideal by considering only the generators of a fixed degree. We investigate cases in which the chopped ideal defines the same finite set of points as the original one-dimensional ideal. The complexity of computing these points from the chopped ideal is governed by the Hilbert function and regularity. We conjecture values for these invariants and prove them in many cases. We show that our conjecture is of practical relevance for symmetric tensor decomposition., Comment: 29 pages, 2 figures, comments are welcome!

We provide a sharp upper bound on the quotient of the rank of an A-hypergeometric system with a three-dimensional torus action by the normalized volume of A; in this case, the upper bound is two., Comment: 9 pages

We compute the $F$-pure threshold of non necessarily principal ideals which satisfy a geometric generic condition about their Newton polygons. We also contribute some evidence in favor of the conjectured equality between the $F$-pure threshold and the log canonical threshold of ideals. These results are obtained by generalizing the theory of splitting polytopes to the case of ideals. As applications of our results we obtain geometric lower bounds for the recently introduced $F$-volume of a collection of ideals., Comment: Comments welcome!

In this paper, we give a purely algebraic approach to the contraction group scheme predicted by Bernstein--Higson--Subag and constructed by Barbasch--Higson--Subag. We also compare quotient schemes of contraction group schemes with other related schemes, equipped with actions of contraction group schemes in the cases of symmetric and $\theta$-stable parabolic subgroups., Comment: The arbitrary base change property of the contraction algebras for smooth algebras was proved (Theorem 2.17 (1)). The connectivity assumption of $G$ in Theorem 1.8 was removed (see also Lemma 2.20). A technical condition in Theorem 4.25 was simplified

We consider Artinian algebras $A$ over a field $\mathsf{k}$, both graded and local algebras. The Lefschetz properties of graded Artinian algebras have been long studied, but more recently the Jordan type invariant of a pair $(\ell,A)$ where $\ell$ is an element of the maximal ideal of $A$, has been introduced. The Jordan type gives the sizes of the Jordan blocks for multiplication by $\ell$ on $A$, and it is a finer invariant than the pair $(\ell,A)$ being strong or weak Lefschetz. The Jordan degree type for a graded Artinian algebra adds to the Jordan type the initial degree of ``strings'' in the decomposition of $A$ as a $\mathsf{k}[\ell]$ module. We here give a brief survey of Jordan type for Artinian algebras, Jordan degree type for graded Artinian algebras, and related invariants for local Artinian algebras, with a focus on recent work and open problems., Comment: 25 pages, comments welcome

In this paper we investigate locally nilpotent derivations on the polynomial algebra in three variables over a field of characteristic zero. We introduce an iterating construction giving all locally nilpotent derivations of rank $2$. This construction allows to get examples of non-triangularizable locally nilpotent derivations of rank $2$. We also show that the well-known example of a locally nilpotent derivation of rank $3$, given by Freudenburg, is a member of a large family of new examples of rank $3$ locally nilpotent derivations. Our approach is based on considering all locally nilpotent derivations commuting with a given one. We obtain a characterization of locally nilpotent derivations with a given rank in terms of sets of commuting locally nilpotent derivations.

$F$-polynomials are integer coefficient polynomials encoding the mutations of cluster variables inside a cluster algebra. In this article, we study the $F$-polynomials associated with the action of Donaldson-Thomas transformations on cluster variables. For acyclic quivers, quivers of surface types, and quivers associated with triples of flags, we give explicit descriptions of their Donaldson-Thomas $F$-polynomials in terms of generating functions for ideals inside a labeled poset. We also describe the combinatorial procedure needed to modify these labeled posets to obtain Donaldson-Thomas $F$-polynomials for full subquivers and triangular extensions., Comment: 27 pages, 19 figures

Samuel conjectured in 1961 that a (Noetherian) local complete intersection ring which is a UFD in codimension at most three is itself a UFD. It is said that Grothendieck invented local cohomology to prove this fact. Following the philosophy that a UFD is nothing else than a Krull domain (that is, a normal domain, in the Noetherian case) with trivial divisor class group, we take a closer look at the Samuel--Grothendieck Theorem and prove the following generalization: Let $A$ be a local Cohen-Macaulay ring. (i) $A$ is a Krull domain if and only if $A$ is a Krull domain in codimension at most $1$. (ii) Suppose that $A$ is a normal domain and a complete intersection. Then the divisor class group of $A$ is a subgroup of the projective limit of the divisor class groups of the localizations $A_p$, where $p$ runs through all prime ideals of height at most $3$ in $A$. We use this fact to describe for an integral separated Noetherian scheme $X$ the gap between the groups of Weil and Cartier divisors, generalizing the classical result that these two concepts coincide if $X$ is locally a UFD.

We prove that a quasi-isomorphism $f : A \to B$ between commutative DG rings, where $B$ admits a divided power structure, can be factored as $f = \tilde{f} \circ e$, where $e : A \to \tilde{B}$ is a split injective quasi-isomorphism, and $\tilde{f} : \tilde{B} \to B$ is a surjective quasi-isomorphism. This result is used in our work on a DG approach to the cotangent complex, and our work on the derived category of commutative DG rings., Comment: 8 pages

Let $K$ be a field of characteristic zero. We deal with the algebraic closure of the field of fractions of the ring of formal power series $K[[x_1,\ldots,x_r]]$, $r\geq 2$. More precisely, we view the latter as a subfield of an iterated Puiseux series field $\mathcal{K}_r$. On the one hand, given $y_0\in \mathcal{K}_r$ which is algebraic, we provide an algorithm that reconstructs the space of all polynomials which annihilates $y_0$ up to a certain order (arbitrarily high). On the other hand, given a polynomial $P\in K[[x_1,\ldots,x_r]][y]$ with simple roots, we derive a closed form formula for the coefficients of a root $y_0$ in terms of the coefficients of $P$ and a fixed initial part of $y_0$., Comment: 74 pages, 6 figures. arXiv admin note: text overlap with arXiv:1702.03709

We provide a complete description of the ideal that serves as the resultant ideal for n univariate polynomials of degree d. We in particular describe a set of generators of this resultant ideal arising as maximal minors of a set of cascading matrices formed from the coefficients of the polynomials, generalising the classical Sylvester resultant of two polynomials., Comment: 9 pages; v2: added some classical references and a new result on projective scheme structure, communicated by Jan Stevens

Given $k \leq 6$ points $(x_i,y_i) \in \mathbb{P}^2 \times \mathbb{P}^2$, we characterize rank deficiency of the $k \times 9$ matrix $Z_k$ with rows $x_i^\top \otimes y_i^\top$ in terms of the geometry of the point configurations $\{x_i\}$ and $\{y_i\}$. While this question comes from computer vision the answer relies on tools from classical algebraic geometry: For $k \leq 5$, the geometry of the rank-drop locus is characterized by cross-ratios and basic (projective) geometry of point configurations. For the case $k=6$ the rank-drop locus is captured by the classical theory of cubic surfaces.