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We study Hilbert functions, Lefschetz properties, and Jordan type of Artinian Gorenstein algebras associated to Perazzo hypersurfaces in projective space. The main focus lies on Perazzo threefolds, for which we prove that the Hilbert functions are always unimodal. Further we prove that the Hilbert function determines whether the algebra is weak Lefschetz, and we characterize those Hilbert functions for which the weak Lefschetz property holds. By example, we verify that the Hilbert functions of Perazzo fourfolds are not always unimodal. In the particular case of Perazzo threefolds with the smallest possible Hilbert function, we give a description of the possible Jordan types for multiplication by any linear form., Comment: 23 pages

We give a short proof that any smooth (means formally smooth and finitely presented) homomorphism of rings can be obtained by base change from a smooth homomorphism of noetherian rings. Together with the elegant short proof by J. Conde-Lago that smooth homomorphisms of noetherian rings are flat, this gives a short and elementary proof of the theorem of Grothendieck that all smooth homomorphisms are flat., Comment: 2 pages

The Quot scheme of points $\mathrm{Quot}_{d,n}(X)$ on a variety $X$ over a field $k$ parametrizes quotient sheaves of $\mathcal{O}_X^{\oplus d}$ of zero-dimensional support and length $n$. It is a rank-$d$ generalization of the Hilbert scheme of $n$ points. When $X$ is a reduced curve with only the cusp singularity $\{x^2=y^3\}$ and $d\geq 0$ is fixed, the generating series for the motives of $\mathrm{Quot}_{d,n}(X)$ in the Grothendieck ring of varieties is studied via Gr\"obner bases, and shown to be rational. Moreover, the generating series is computed explicitly when $d\leq 3$. The computational results exhibit surprising patterns (despite the fact that the category of finite length coherent modules over a cusp is wild), which not only enable us to conjecture the exact form of the generating series for all $d$, but also suggest a general functional equation whose $d=1$ case is the classical functional equation of the motivic zeta function known for any Gorenstein curve. As another side of the story, Quot schemes are related to the Cohen--Lenstra series. The Cohen--Lenstra series encodes the count of "commuting matrix points'' (or equivalently, coherent modules of finite length) of a variety over a finite field, about which Huang formuated a "rationality'' conjecture for singular curves. We prove a general formula that expresses the Cohen--Lenstra series in terms of the motives of the (punctual) Quot schemes, which together with our main rationality theorem, provides positive evidence for Huang's conjecture for the cusp., Comment: 64 pages, 14 figures, 4 tables; minor changes

We compute the field of rational local unitary invariants for several types of mixed states of two qubits that are of interest in quantum information theory. In particular, we consider locally maximally mixed states, symmetrically mixed states and the full space of mixed states. In each case, we construct explicit geometric quotients and prove that they are always rational. As a consequence, we prove in each case that the field of rational invariants is purely transcendental. All the results are obtained by working over the field of real numbers, employing methods from classical and geometric invariant theory over arbitrary fields of characteristic zero.

We give a geometric interpretation of the Stanley--Reisner correspondence, extend it to schemes, and interpret it in terms of the field of one element.

For a 0-dimensional scheme $\mathbb{X}$ in $\mathbb{P}^n$ over a perfect field $K$, we first embed the homogeneous coordinate ring $R$ into its truncated integral closure $\widetilde{R}$. Then we use the corresponding map from the module of K\"ahler differentials $\Omega^1_{R/K}$ to $\Omega^1_{\widetilde{R}/K}$ to find a formula for the Hilbert polynomial ${\rm HP}(\Omega^1_{R/K})$ and a sharp bound for the regularity index ${\rm ri}(\Omega^1_{R/K})$. Additionally, we extend this to formulas for the Hilbert polynomials ${\rm HP}(\Omega^m_{R/K})$ and bounds for the regularity indices of the higher modules of K\"ahler differentials. Next we derive a new characterization of a weakly curvilinear scheme $\mathbb{X}$ which can be checked without computing a primary decomposition of its homogeneous vanishing ideal. Moreover, we prove precise formulas for the Hilbert polynomial of $\Omega^m_{R/K}$ of a fat point scheme $\mathbb{X}$, extending and settling previous partial results and conjectures. Finally, we characterize uniformity conditions on $\mathbb{X}$ using the Hilbert functions of the K\"ahler differential modules of $\mathbb{X}$ and its subschemes., Comment: 32 pages

We give conditions characterizing equality in the Minkowski inequality for big divisors on a projective variety. Our results draw on the extensive history of research on Minkowski inequalities in algebraic geometry., 27 pages. A reference to a recent preprint has been added

We construct two families of free resolutions that resolve the ideals of certain opposite Schubert varieties restricted to the big open cell. We conjecture that these examples have genericity properties translating to structure theorems for perfect ideals with given Betti numbers, extending the well-known theorem of Buchsbaum and Eisenbud on Gorenstein ideals of codimension three.

Milnor patching diagram is essentially the commutative square of rings, over which gluing of finitely generated projective modules is possible in the strongest sense. Necessary and sufficient conditions for a square to be Milnor patching diagram were studied by Milnor, Beauville-Laszlo and Landsburg. We relate this question to determinant-induced factorization in matrix rings to construct a series of non-surjective Milnor patching diagrams, settling the question of Landsburg, and make a step towards the classification of such examples. Also we consider a possible generalization of the notion of Milnor patching diagram to arbitrary subcategories of modules and obtain a classification result for this setting., Comment: 13 pages

In the present note we construct new families of free plane curves starting from a curve $C$ and adding high order inflectional tangent lines of $C$, lines joining the singularities of the curve $C$, or lines in the tangent cone of some singularities of $C$. These lines $L$ have in common that the intersection $C \cap L$ consists of a small number of points. We introduce the notion of a supersolvable plane curve and conjecture that such curves are always free, as in the known case of line arrangements. Some evidence for this conjecture is given as well, both in terms of a general result in the case of quasi homogeneous singularities and in terms of specific examples. We construct a new example of maximizing curve in degree 8 and the first and unique known example of maximizing curve in degree 9. In the final section, we use a stronger version of a result due to Schenck, Terao and Yoshinaga to construct families of free conic-line arrangements by adding lines to the conic-line arrangements of maximal Tjurina number recently classified by V. Beorchia and R. M. Mir\'o-Roig in arXiv:2303.04665., Comment: v.3: section 7 is new: here we use a stronger version of a result due to Schenck, Terao and Yoshinaga to construct families of free conic-line arrangements by adding lines to the conic-line arrangements of maximal Tjurina number recently classified by V. Beorchia and R. M. Mir\'o-Roig in arXiv:2303.04665

We develop a formalism of unit $F$-modules in the style of Lyubeznik and Emerton-Kisin for rings which have finite $F$-representation type after localization and completion at every prime ideal. As applications, we show that if $R$ is such a ring then the iterated local cohomology modules $H^{n_1}_{I_1} \circ \cdots \circ H^{n_s}_{I_s}(R)$ have finitely many associated primes, and that all local cohomology modules $H^n_I(R / gR)$ have closed support when $g$ is a nonzerodivisor on $R$., Comment: 25 pages, comments appreciated

Abhyankar defined an ideal to be Hilbertian if its Hilbert polynomial coincides with its Hilbert function for all nonnegative integers. In 1984, he proved that the ideal of (r+1)-order minors of a generic p x q matrix is Hilbertian. We give a different proof and a generalization to the Schubert determinantal ideals introduced by Fulton in 1992. Our proof reduces to a simple upper bound for the Castelnuovo-Mumford regularity of these ideals. We further indicate the pervasiveness of the Hilbertian property in Schubert geometry., Comment: With thanks to Allen Knutson and Jenna Rajchgot for feedback about v1, v2 contains additional results and background material. 12 pages

Valuated term orders are studied for the purposes of Gr\"{o}bner theory over fields with valuation. The points of a usual tropical variety correspond to certain valuated terms preorders. Generalizing both of these, the set of all ``well-behaved'' valuated term preorders is canonically in bijection with the points of a space introduced in our previous work on tropical adic geometry. In this paper we interpret these points geometrically by explicitly characterizing them in terms of classical polyhedral geometry. This characterization gives a bijection with equivalence classes of flags of polyhedra as well as a bijection with a class of prime filters on a lattice of polyhedral sets. The first of these also classifies valuated term orders. The second bijection is of the same flavor as the bijections from [van der Put and Schneider, 1995] in non-archimedean analytic geometry and indicates that the results of that paper may have analogues in tropical adic geometry., Comment: This was originally the second half of arXiv:2209.15116v2

This work concerns generators for the bounded derived category of coherent sheaves over a noetherian scheme $X$ of prime characteristic. The main result is that when the Frobenius map on $X$ is finite, for any compact generator $G$ of $\mathsf{D}(X)$ the Frobenius pushforward $F ^e_*G$ generates the bounded derived category whenever $p^e$ is larger than the codepth of $X$, an invariant that is a measure of the singularity of $X$. The conclusion holds for all positive integers $e$ when $X$ is locally complete intersection. The question of when one can take $G=\mathcal{O}_X$ is also investigated. For smooth projective complete intersections it reduces to a question of generation of the Kuznetsov component., Comment: 31 pages. Minor edits

We study commutative Cohen-Macaulay rings whose Cohen-Macaulay representation theory are controlled by representations of quivers, which we call hereditary representation type. Based on tilting theory and cluster tilting theory, we construct some commutative Cohen-Macaulay rings of hereditary representation type. First we give a general existence theorem of cluster tilting module or non-commutative crepant resolutions on the Segre product of two commutative Gorenstein rings whenever each factor has such an object. As an application we obtain three examples of Gorenstein rings of hereditary representation type coming from Segre products of polynomial rings. Next we introduce extended numerical semigroup rings which generalize numerical semigroup rings and form a class of one-dimensional Cohen-Macaulay non-domains, and among them we provide one family of Gorenstein rings of hereditary representation type. Furthermore, we discuss a $4$-dimensional non-Gorenstein Cohen-Macaulay ring whose representations are still controlled by a finite dimensional hereditary algebra. We show that it has a unique $2$-cluster tilting object, and give a complete classification of rigid Cohen-Macaulay modules, which turns out to be only finitely many., Comment: 35 pages

We discuss the space $VPS(Q,H)$ of ideals with Hilbert function $H=(1,n,n, \ldots )$ that are apolar to a full rank quadric $Q$. We prove that its components of saturated ideals are closely related to the locus of Gorenstein algebras and to the Slip component in border apolarity. We also point out an important error in~\cite{RS} and provide the necessary corrections.

We analyze adjunction and inversion of adjunction for the $F$-purity of divisor pairs in characteristic $p > 0$. In this vein, we give a complete answer for principal divisors under $\mathbb{Q}$-Gorenstein assumptions but without divisibility restrictions on the index. We also give a detailed analysis relating the $F$-purity of the pairs $(R,\Delta + D)$ and that of $(R_D, \text{Diff}_D(\Delta))$ motivated by Kawakita's log canonical inversion of adjunction via reduction to prime characteristic., Comment: 21 pages, comments welcome

The theorem of Duistermaat and Van der Kallen from 1998 proved the first case of the Mathieu conjecture. Using the theory of Mathieu-Zhao spaces, we can reformulate this theorem as $\operatorname{Ker} L$ is a Mathieu-Zhao space where $L$ is the linear map \begin{align*} L\colon {\bf C}[X_1,\ldots,X_n,X_1^{-1},\ldots,X_n^{-1}] \to C,\ f \mapsto f_0\end{align*}. In this paper, we generalize this result (for $n = 1$) to all non-trivial linear maps $L\colon C[X,X^{-1}] \to C$ such that $\{X^n \mid |n|\geq N\} \subset \operatorname{Ker} L$ for some $N \geq 1$.

It is shown that in each dimension there are only finitely many projective algebraic varieties occurring as a moduli space of representations of a quiver with a dimension vector that satisfies certain simple constraints., Comment: v2: Abstract corrected, minor edits

In this note we find an alternate characterization of the multiplier ideal of a normal variety $X$, as defined by de Fernex-Hacon, by considering maps $\pi_*\omega_Y\to\mathcal{O}_X$ where $\pi:Y\to X$ ranges over all regular alterations. As a corollary to this result, we give a derived splinter characterization of klt singularities, akin to the characterization of rational singularities given by Kov\'acs and Bhatt. We also give an analogous description of the test ideal in characteristic $p>2$ as a corollary to a result of Epstein-Schwede., Comment: 15 pages, comments welcome!

Let $[Z]\in\text{Hilb}^d \mathbb A^3$ be a zero-dimensional subscheme of the affine three-dimensional complex space of length $d>0$. Okounkov and Pandharipande have conjectured that the dimension of the tangent space of $\text{Hilb}^d \mathbb A^3$ at $[Z]$ and $d$ have have the same parity. The conjecture was proven by Maulik, Nekrasov, Okounkov and Pandharipande for points $[Z]$ defined by monomial ideals and very recently by Ramkumar and Sammartano for homogeneous ideals. In this paper we exhibit a family of zero-dimensional schemes in $\text{Hilb}^{12} \mathbb A^3$, which disproves the conjecture in the general non-homogeneous case., 11 pages. Comments are welcome

Elliptic curves are fundamental objects in number theory and algebraic geometry, whose points over a field form an abelian group under a geometric addition law. Any elliptic curve over a field admits a Weierstrass model, but prior formal proofs that the addition law is associative in this model involve either advanced algebraic geometry or tedious computation, especially in characteristic two. We formalise in the Lean theorem prover, the type of nonsingular points of a Weierstrass curve over a field of any characteristic and a purely algebraic proof that it forms an abelian group., Comment: Submitted to 14th International Conference on Interactive Theorem Proving (ITP 2023), source code in https://github.com/alreadydone/mathlib/tree/associativity

The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by $\Bbbk[K_{r_1,\ldots,r_n}]$, where $1 \leq r_1 \leq \cdots \leq r_n$. More concretely, we prove that the class group of $\Bbbk[K_{r_1,\ldots,r_n}]$ is isomorphic to $\mathbb{Z}^n$ if $n =3$ with $r_1 \geq 2$ or $n \geq 4$, while it turns out that the excluded cases can be deduced into Hibi rings. The second goal is to investigate the special class of divisorial ideals of $\Bbbk[K_{r_1,\ldots,r_n}]$, called conic divisorial ideals. We describe conic divisorial ideals for certain $K_{r_1,\ldots,r_n}$ including all cases where $\Bbbk[K_{r_1,\ldots,r_n}]$ is Gorenstein. Finally, we give a non-commutative crepant resolution (NCCR) of $\Bbbk[K_{r_1,\ldots,r_n}]$ in the case where it is Gorenstein., 21 pages

The containment problem for symbolic and ordinary powers of ideals asks for what values of $a$ and $b$ we have $I^{(a)} \subseteq I^b$. Over a regular ring, a result by Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede partially answers this question, but the containments it provides are not always best possible. In particular, a tighter containment conjectured by Harbourne has been shown to hold for interesting classes of ideals - although it does not hold in general. In this paper, we develop a Fedder (respectively, Glassbrenner) type criterion for $F$-purity (respectively, strong $F$-regularity) for ideals of finite projective dimension over $F$-finite Gorenstein rings and use our criteria to extend the prime characteristic results of Grifo-Huneke to singular ambient rings. For ideals of infinite projective dimension, we prove that a variation of the containment still holds, in the spirit of work by Hochster-Huneke and Takagi., Comment: Final version

We prove that seminormality of cut polytopes is equivalent to normality. This settles two conjectures regarding seminormality of cut polytopes.

Tropical ideals are a class of ideals in the tropical polynomial semiring that combinatorially abstracts the possible collections of supports of all polynomials in an ideal over a field. We study zero-dimensional tropical ideals I with Boolean coefficients in which all underlying matroids are paving matroids, or equivalently, in which all polynomials of minimal support have support of size deg(I) or deg(I)+1 -- we call them paving tropical ideals. We show that paving tropical ideals of degree d+1 are in bijection with $\mathbb Z^n$-invariant d-partitions of $\mathbb Z^n$. This implies that zero-dimensional tropical ideals of degree 3 with Boolean coefficients are in bijection with $\mathbb Z^n$-invariant 2-partitions of quotient groups of the form $\mathbb Z^n/L$. We provide several applications of these techniques, including a construction of uncountably many zero-dimensional degree-3 tropical ideals in one variable with Boolean coefficients, and new examples of non-realizable zero-dimensional tropical ideals., Comment: 13 pages

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

Finite generation of the symbolic Rees ring of a space monomial prime ideal of a 3-dimensional weighted polynomial ring is a very interesting problem. Negative curves play important roles in finite generation of these rings. We are interested in the structure of the negative curve. We shall prove that negative curves are rational in many cases. We also see that the Cox ring of the blow-up of a toric variety at the point (1,1,...,1) coincides with the extended symbolic Rees ring of an ideal of a polynomial ring. For example, Roberts' second counterexample to Cowsik's question (and Hilbert's 14th problem) coincides with the Cox ring of some normal projective variety., Comment: 24 pages

We show that the number of rational points of a subgroup inside a toric variety over a finite field defined by a homogeneous lattice ideal can be computed via Smith normal form of the matrix whose columns constitute a basis of the lattice. This generalizes and yields a concise toric geometric proof of the same fact proven purely algebraically by Lopez and Villarreal for the case of a projective space and a standard homogeneous lattice ideal of dimension one. We also prove a Nullstellensatz type theorem over a finite field establishing a one to one correspondence between subgroups of the dense split torus and certain homogeneous lattice ideals. As application, we compute the main parameters of generalized toric codes on subgroups of the torus of Hirzebruch surfaces, generalizing the existing literature., Comment: 26 pages, 5 figures, supported by TUBITAK 119F177, to appear in Finite Fields and their Applications

In this note, we study Togliatti systems generated by invariants of the dihedral group $D_{2d}$ acting on $k[x_{0},x_{1},x_{2}]$. This leads to the first family of non monomial Togliatti systems, which we call $GT-$systems with group $D_{2d}$. We study their associated varieties $S_{D_{2d}}$, called $GT-$surfaces with group $D_{2d}$. We prove that they are arithmetically Cohen-Macaulay surfaces whose homogeneous ideal, $I(S_{D_{2d}})$, is minimally generated by quadrics and we find a minimal free resolution of $I(S_{D_{2d}})$., Comment: To appear in Israel Journal of Mathematics

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

For an essentially small triangulated category $\mathcal{T}$, we introduce the notion of prime thick subcategories and define the spectrum of $\mathcal{T}$, which shares the basic properties with the spectrum of a tensor triangulated category introduced by Balmer. We mainly focus on triangulated categories that appear in algebraic geometry such as the derived and the singularity categories of a noetherian scheme $X$. We prove that certain classes of thick subcategories are prime thick subcategories of these triangulated categories. Furthermore, we use this result to show that certain subspaces of $X$ are embedded into their spectra as topological spaces., 20 pages

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

Let $k$ be a perfect field of characteristic $\neq 2$. We prove that the Schmidt rank (also known as strength) of a quartic polynomial $f$ over $k$ is bounded above in terms of only the Schmidt rank of $f$ over $\overline{k}$, an algebraic closure of $k$., 13 pages. arXiv admin note: text overlap with arXiv:2107.08080

In this article, we study Hilbert Series of non-Cohen-Maculay tangent cones for some 4-generated pseudo symmetric monomial curves. We show that the Hilbert Function is nondecreasing by explicitly computing it. We also compute standard bases of these toric ideals.

We describe the twisted $K$-polynomial of multiplicity-free varieties in a multiprojective setting. More precisely, for multiplicity-free varieties, we show that the support of the twisted $K$-polynomial is a generalized polymatroid. As applications, we show that the support of the M\"obius function of a linear polymatroid is a generalized polymatroid, and we settle a conjecture of Monical, Tokcan and Yong regarding Grothendieck polynomials for the case of zero-one Schubert polynomials.

Let $K$ be a perfectoid field with pseudo-uniformizer $\pi$. We adapt an argument of Du to show that the perfectoid Tate algebra $K\langle x^{1 / p^{\infty}} \rangle$ has an uncountable chain of distinct prime ideals. First, we conceptualize Du's argument, defining the notion of a 'Newton polygon formalism' on a ring. We prove a version of Du's theorem in the prescence of a sufficiently nondiscrete Newton polygon formalism. Then, we apply our framework to the perfectoid Tate algebra via a "nonstandard" Newton polygon formalism (roughly, the roles of the series variable $x$ and the pseudo-uniformizer $\pi$ are switched). We conclude a similar statement for multivatiate perfectoid Tate algebras using the one-variable case. We also answer a question of Heitmann, showing that if $R$ is a complete local noetherian domain of mixed characteristic $(0,p)$, the $p$-adic completion of it's absolute integral closure $R^{+}$ has uncountable Krull dimension., Comment: 15 pages, 2 figures, comments welcome

This paper develops two theories, the geometric theory of derived Grassmannians (and flag schemes) and the algebraic theory of derived Schur (and Weyl) functors, and establishes their connection, a derived generalization of the Borel-Weil-Bott theorem. More specifically: (1) The theory of derived Grassmannians and flag schemes is the natural extension of the theory of derived projectivizations [arXiv:2202.11636] and generalizes Grothendieck's theory of Grassmannians and flag schemes of sheaves to the case of complexes. We establish their fundamental properties and study various natural morphisms among them. (2) The theory of derived Schur and Weyl functors extends the classical theory of Schur and Weyl module functors studied in $\mathrm{GL}_n(\mathbb{Z})$-representation theory to the case of complexes. We show that these functors have excellent functorial properties and satisfy derived generalizations of classical formulae such as Cauchy's decomposition formula, direct-sum decomposition formula and Littlewood-Richardson rule. We also generalize various results from the case of derived symmetric powers to derived Schur functors, such as Illusie-Lurie's d\'ecalage isomorphism. (3) These two theories are connected by a derived version of the Borel-Weil-Bott theorem, which generalizes the classical Borel-Weil-Bott theorem and calculates the derived pushforwards of tautological perfect complexes on derived flag schemes in terms of derived Schur functors when the complexes have perfect-amplitude $\le 1$ and positive ranks., Comment: 99 pages. Comments welcome! v2: References updated, and a typo in the formulation of d\'ecalage isomorphisms corrected

The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semialgebraic set $S$ and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter $\epsilon$ measuring the non-vanishing of the positive function, the constant $c$ and exponent $L$ of a {\L}ojasiewicz inequality for the semi-algebraic distance function associated to the inequalities $g = (g_1 ,. .. , g_r)$ defining $S$. They are polynomial in $c$ and $\epsilon^{-1}$ with an exponent depending only on $L$. We analyse in details the {\L}ojasiewicz inequality when the defining inequalties g satisfy the Constraint Qualification Condition. We show that, in this case, the {\L}ojasiewicz exponent $L$ is 1 and we relate the {\L}ojasiewicz constant $c$ with the distance of $g$ to the set of singular systems.

Let $S$ be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let $I$ be the determinantal ideal of maximal minors or $\operatorname{Pf}$ the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu--Weyman--Witt to determine the $S$-module structures of $\operatorname{Ext}^j_S(S/I^t, S)$ and $\operatorname{Ext}^j_S(S/\operatorname{Pf}^t, S)$, from which we get the degrees of generators of these $\operatorname{Ext}$ modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang on the socle degrees of local cohomology modules of the form $H^j_\mathfrak{m}(S/I^t)$., Comments welcome!

We give a construction of generalized cluster varieties and generalized cluster scattering diagrams for reciprocal generalized cluster algebras, the latter of which were defined by Chekhov and Shapiro. These constructions are analogous to the structures given for ordinary cluster algebras in the work of Gross, Hacking, Keel, and Kontsevich. As a consequence of these constructions, we are also able to construct theta functions for generalized cluster algebras, again in the reciprocal case, and demonstrate a number of their structural properties., Comment: Updated to reflect reviewers' comments and to use the preferred journal formatting

Let $G\subset GL_n(k)$ be a finite subgroup and $k[x_1,\dots, x_n]^G\subset k[x_1,\dots, x_n]$ its ring of invariants. We show that, in many cases, the automorphism group of $k[x_1,\dots, x_n]^G$ is $k^\times$. Version 2: Incorporates parts of arXiv:2210.16265.

Let $X$ be a generic determinantal affine variety over a perfect field of characteristic $p \geq 0$ and $P \subset X$ be a standard prime divisor generator of $\mathrm{Cl}(X) \cong \mathbb{Z}$. We prove that the pair $(X,P)$ is purely $F$-regular if $p>0$ and so that $(X,P)$ is purely log terminal (PLT) if $p=0$ and $(X,P)$ is log $\mathbb{Q}$-Gorenstein. In general, using recent results of Z. Zhuang and S. Lyu, we show that $(X,P)$ is of PLT-type, i.e. there is a $\mathbb{Q}$-divisor $\Delta$ with coefficients in $[0,1)$ such that $(X,P+\Delta)$ is PLT., Comment: 23 pages, minor corrections, final version, accepted for publication at the Bollettino dell'Unione Matematica Italiana

We study the Hadamard product of two varieties $V$ and $W$, with particular attention to the situation when one or both of $V$ and $W$ is a binomial variety. The main result of this paper shows that when $V$ and $W$ are both binomial varieties, and the binomials that define $V$ and $W$ have the same binomial exponents, then the defining equations of $V \star W$ can be computed explicitly and directly from the defining equations of $V$ and $W$. This result recovers known results about Hadamard products of binomial hypersurfaces and toric varieties. Moreover, as an application of our main result, we describe a relationship between the Hadamard product of the toric ideal $I_G$ of a graph $G$ and the toric ideal $I_H$ of a subgraph $H$ of $G$. We also derive results about algebraic invariants of Hadamard products: assuming $V$ and $W$ are binomial with the same exponents, we show that $\text{deg}(V\star W) = \text{deg}(V)=\text{deg}(W)$ and $\dim(V\star W) = \dim(V)=\dim(W)$. Finally, given any (not necessarily binomial) projective variety $V$ and a point $p \in \mathbb{P}^n \setminus \mathbb{V}(x_0x_1\cdots x_n)$, subject to some additional minor hypotheses, we find an explicit binomial variety that describes all the points $q$ that satisfy $p \star V = q\star V$., 24 pages, comments welcome

In this paper, using ultra-Frobenii, we introduce a variant of Schoutens' non-standard tight closure, ultra-tight closure, on ideals of a local domain $R$ essentially of finite type over $\mathbb{C}$. We prove that the ultra-test ideal $\tau_{\rm u}(R,\mathfrak{a}^t)$, the annihilator ideal of all ultra-tight closure relations of $R$, coincides with the multiplier ideal $\mathcal{J}(\operatorname{Spec} R,\mathfrak{a}^t)$ if $R$ is normal $\mathbb{Q}$-Gorenstein. As an application, we study a behavior of multiplier ideals under pure ring extensions., Comment: 15 pages. The proof of Thm 5.18 revised since $F^\epsilon_*S_\infty$ may not be isomorphic to $F^\epsilon_* R_\infty\otimes_R S$ in codim one even if $F_*^{e_p}S_p$ is isomorphic to $F_*^{e_p}R_p\otimes_{R_p}S_p$ in codim one for almost all $p$

Let $(R,\mathfrak{m}_R,k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero. R. Berger conjectured that $R$ is regular if and only if the universally finite module of differentials $\Omega_R$ is torsion free. When $R$ is a domain, we prove the conjecture in several cases. Our techniques are primarily reliant on making use of the valuation semi-group of $R$. First, we establish a method of verifying the conjecture by analyzing the valuation semi-group of $R$ and orders of units of the integral closure of $R$. We also prove the conjecture in the case when certain monomials are missing from the monomial support of the defining ideal of $R$. These monomials are based on the smallest power of $\mathfrak{m}_R$ that is contained within the conductor ideal. This also generalizes a previous result of Corti\~nas, Geller and Weibel., Comment: 18 Pages. Comments are welcome

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

An excellent ring of prime characteristic for which the Frobenius map is pure is also Frobenius split in many commonly occurring situations in positive characteristic commutative algebra and algebraic geometry. However, using a fundamental construction from rigid geometry, we show that excellent $F$-pure rings of prime characteristic are not Frobenius split in general, even for Euclidean domains. Our construction uses the existence of a complete non-Archimedean field $k$ of characteristic $p$ with no nonzero continuous $k$-linear maps $k^{1/p} \to k$. An explicit example of such a field is given based on ideas of Gabber, and may be of independent interest. Our examples settle a long-standing open question in the theory of $F$-singularities whose origin can be traced back to when Hochster and Roberts introduced the notion of $F$-purity. The excellent Euclidean domains we construct also admit no nonzero $R$-linear maps $R^{1/p} \rightarrow R$. These are the first examples that illustrate that $F$-purity and Frobenius splitting define different classes of singularities for excellent domains, and are also the first examples of excellent domains with no nonzero $p^{-1}$-linear maps. The latter is particularly interesting from the perspective of the theory of test ideals., Comment: 17 pages. Comments welcome! v3: Minor changes. v2: Fixed typos

We study varieties associated to hypergraphs from the point of view of projective geometry and matroid theory. We describe their decompositions into matroid varieties, which may be reducible and can have arbitrary singularities by the Mn\"ev--Sturmfels universality theorem. We focus on various families of hypergraph varieties for which we explicitly compute an irredundant irreducible decomposition. Our main findings in this direction are threefold: (1) we describe minimal matroids of such hypergraphs; (2) we prove that the varieties of these matroids are irreducible and their union is the hypergraph variety; and (3) we show that every such matroid is realizable over real numbers. As corollaries, we give conceptual decompositions of various, previously-studied, varieties associated with graphs, hypergraphs, and adjacent minors of generic matrices. In particular, our decomposition strategy gives immediate matroid interpretations of the irreducible components of multiple families of varieties associated to conditional independence (CI) models in statistical theory and unravels their symmetric structures which hugely simplifies the computations., Comment: Final version to appear in International Mathematics Research Notices (IMRN)

The ACC conjecture for local volumes predicts that the set of local volumes of klt singularities $x\in (X,\Delta)$ satisfies the ACC if the coefficients of $\Delta$ belong to a DCC set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of $\delta$-plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities., Comment: Final version, 50 pages, to appear in J. Algebraic Geom., appendix A will not appear in the journal version, comments are very welcome!