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Classically, the Bezout matrix or simply Bezoutian of two polynomials is used to locate the roots of the polynomial and, in particular, test for stability. In this paper, we develop the theory of Bezoutians on real Riemann surfaces of dividing type. The main result connects the signature of the Bezoutian of two real meromorphic functions to the topological data of their quotient, which can be seen as the generalization of the classical Cauchy index. As an application, we propose a method to count the number of zeroes of a polynomial in a quadrature domain using the inertia of the Bezoutian. We provide examples of our method in the case of simply connected quadrature domains., Comment: 23 pages, 2 figures. Comments are welcome

Given an ample action of an inverse semigroup on a locally compact and Hausdorff topological space, we study the ideal structure of the crossed product algebra associated with it. By developing a theory of induced ideals, we manage to prove that every ideal in the crossed product algebra may be obtained as the intersection of ideals induced from isotropy group algebras. This can be interpreted as an algebraic version of the Effros-Hahn conjecture. Finally, as an application of our result, we study the ideal structure of a Steinberg algebra associated with an ample groupoid by interpreting it as an inverse semigroup crossed product algebra., Comment: This paper has resulted from the author's work as a PhD student in Florianopolis under the advice of professor Ruy Exel. arXiv admin note: substantial text overlap with arXiv:1605.07540 by other authors

We introduce higher dimensional analogues of the Nakayama algebras from the viewpoint of Iyama's higher Auslander--Reiten theory. More precisely, for each Nakayama algebra $A$ and each positive integer $d$, we construct a finite dimensional algebra $A^{(d)}$ having a distinguished $d$-cluster-tilting $A^{(d)}$-module whose endomorphism algebra is a higher dimensional analogue of the Auslander algebra of $A$. We also construct higher dimensional analogues of the mesh category of type $\mathbb{ZA}_\infty$ and the tubes., Comment: v5: 50 pages, further minor corrections following referee report. With an appendix by the second named author and Chrysostomos Psaroudakis and an appendix by Sondre Kvamme

Let ( R , m ) be a Noetherian local ring, and let M be a finitely generated R-module of dimension d. We prove that the set { l ( M / I M ) e ( I , M ) } I = m is bounded below by 1 / d ! e ( R ‾ ) where R ‾ = R / Ann ( M ) . Moreover, when M ˆ is equidimensional, this set is bounded above by a finite constant depending only on M. The lower bound extends a classical inequality of Lech, and the upper bound answers a question of Stuckrad–Vogel in the affirmative. As an application, we obtain results on uniform behavior of the lengths of Koszul homology modules.

We introduce a new family of algebras, called Serre-formal algebras. They are Iwanaga–Gorenstein algebras for which applying any power of the Serre functor on any indecomposable projective module, the result remains a stalk complex. Typical examples are given by (higher) hereditary algebras and self-injective algebras; it turns out that other interesting algebras such as (higher) canonical algebras are also Serre-formal. Starting from a Serre-formal algebra, we consider a series of algebras – called the replicated algebras – given by certain subquotients of its repetitive algebra. We calculate the self-injective dimension and dominant dimension of all such replicated algebras and determine which of them are minimal Auslander–Gorenstein, i.e. when the two dimensions are finite and equal to each other. In particular, we show that there exist infinitely many minimal Auslander–Gorenstein algebras in such a series if, and only if, the Serre-formal algebra is twisted fractionally Calabi–Yau. We apply these results to a construction of algebras from Yamagata [29] , called SGC extensions, given by iteratively taking the endomorphism ring of the smallest generator-cogenerator. We give a sufficient condition so that the SGC extensions and replicated algebras coincide. Consequently, in such a case, we obtain explicit formulae for the self-injective dimension and dominant dimension of the SGC extension algebras.

We show that, in the context of Reverse Mathematics, WK L 0 (Weak Konig's Lemma) implies the statement AR T 0 which says that every Artinian ring is Noetherian, over RC A 0 (Recursive Comprehension Axiom). To achieve this goal, we prove a general Computable Full Structure Theorem for computable Artinian rings similar to the classical version found in most Algebra texts.

Matrix Schubert varieties are the closures of the orbits of $B\times B$ acting on all $n\times n$ matrices, where $B$ is the group of invertible lower triangular matrices. Extending work of Fulton, Knutson and Miller identified a Gr\"obner basis for the prime ideals of these varieties. They also showed that the corresponding initial ideals are Stanley-Reisner ideals of shellable simplicial complexes, and derived a related primary decomposition in terms of reduced pipe dreams. These results lead to a geometric proof of the Billey-Jockusch-Stanley formula for a Schubert polynomial, among many other applications. We define skew-symmetric matrix Schubert varieties to be the nonempty intersections of matrix Schubert varieties with the subspace of skew-symmetric matrices. In analogy with Knutson and Miller's work, we describe a natural generating set for the prime ideals of these varieties. We then compute a related Gr\"obner basis. Using these results, we identify a primary decomposition for the corresponding initial ideals involving certain fpf-involution pipe dreams. We show that these initial ideals are likewise the Stanley-Reisner ideals of shellable simplicial complexes. As an application, we give a geometric proof of an explicit generating function for symplectic Grothendieck polynomials. Our methods differ from Knutson and Miller's and can be used to give new proofs of some of their results, as we explain at the end of this article., Comment: 45 pages; v2: fixed several typos; v3: minor corrections, updated references, final version

We give a long exact sequence for the homology of a graded atomic lattice equipped with a sheaf of modules, in terms of the deleted and restricted lattices. This is then used to compute the homology of the arrangement lattice of a hyperplane arrangement equipped with the natural sheaf. This generalises an old result of Lusztig.

The purpose of this paper is to establish a Castelnuovo-Mumford regularity bound for threefolds with mild singularities. Let $X$ be a non-degenerate normal projective threefold in $\mathbb{P}^r$ of degree $d$ and codimension $e$. We prove that if $X$ has rational singularities, then $\text{reg}(X) \leq d-e+2$. Our bound is very close to a sharp bound conjectured by Eisenbud-Goto. When $e=2$ and $X$ has Cohen-Macaulay Du Bois singularities, we obtain the conjectured bound $\text{reg}(X) \leq d-1$, and we also classify the extremal cases. To achieve these results, we bound the regularity of fibers of a generic projection of $X$ by using Loewy length, and also bound the dimension of the varieties swept out by secant lines through the singular locus of $X$., Comment: 17 pages, to appear in Adv. Math

Let $S$ be a polynomial ring over a field and $I\subseteq S$ a homogeneous ideal containing a regular sequence of forms of degrees $d_1, \ldots, d_c$. In this paper we prove the Lex-plus-powers Conjecture when the field has characteristic 0 for all regular sequences such that $d_i \geq \sum_{j=1}^{i-1} (d_j-1)+1$ for each $i$; that is, we show that the Betti table of $I$ is bounded above by the Betti table of the lex-plus-powers ideal of $I$. As an application, when the characteristic is 0, we obtain bounds for the Betti numbers of any homogeneous ideal containing a regular sequence of known degrees, which are sharper than the previously known ones from the Bigatti-Hulett-Pardue Theorem., To appear in Advances in Mathematics

We consider the (graded) Matlis dual D ( M ) of a graded D -module M over the polynomial ring R = k [ x 1 , … , x n ] (k is a field of characteristic zero), and show that it can be given a structure of D -module in such a way that, whenever dim k ⁡ H d R i ( M ) is finite, then H d R i ( M ) is k-dual to H d R n − i ( D ( M ) ) . As a consequence, we show that if M is a graded D -module such that H d R n ( M ) is a finite-dimensional k-space, then dim k ⁡ ( H d R n ( M ) ) is the maximal integer s for which there exists a surjective D -linear homomorphism M → E s , where E is the top local cohomology module H ( x 1 , … , x n ) n ( R ) . This extends a recent result of Hartshorne and Polini on formal power series rings to the case of polynomial rings; we also apply the same circle of ideas to provide an alternate proof of their result. When M is a finitely generated graded D -module such that dim k ⁡ H d R i ( M ) is finite, we generalize the above result further, showing that H d R n − i ( M ) is k-dual to Ext D i ( M , E ) .

Let G be a finite p-subgroup of GL(V), where p = char(F), and V is finite-dimensional over the field F. Let S(V) be the symmetric algebra of V, S(V)^G the subring of G-invariants, and V* the F-dual space of V. The following presents our solution to the above question. Theorem A. Suppose dim(V)=3. Then S(V)^G is a polynomial ring if and only if G is generated by transvections. Theorem B. Suppose dim(V) > 3. Then S(V)^G is a polynomial ring if and only if: (1) S(V)^{G_U} is a polynomial ring for each subspace U of V* with dim(U)=2, where G_U = {g in G | g(u) = u, for all u in U}, and (2) S(V)^G is Cohen-Macaulay. Alternatively, (1) can be replaced by the equivalent condition: (3) dim (non-smooth locus of S(V)^G) < 2., 31 pages

We give a classification of all exact structures on a given idempotent complete additive category. Using this, we investigate the structure of an exact category with finitely many indecomposables. We show that the relation of the Grothendieck group of such a category is generated by AR conflations. Moreover, we obtain an explicit classification of (1) Gorenstein-projective-finite Iwanaga-Gorenstein algebras, (2) Cohen-Macaulay-finite orders, and more generally, (3) cotilting modules $U$ with $^\perp U$ of finite type. In the appendix, we develop the AR theory of exact categories over a noetherian complete local ring, and relate the existence of AR conflations to the AR duality and dualizing varieties., 27 pages, Final version

In this paper, we study the asymptotic behavior of the Hermitian–Yang–Mills flow on a reflexive sheaf. We prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the Harder–Narasimhan–Seshadri filtration, this answers a question by Bando and Siu.

We generalize the notions of n-cluster tilting subcategories and τ-selfinjective algebras into n-precluster tilting subcategories and τ n -selfinjective algebras, where we show that a subcategory naturally associated to n-precluster tilting subcategories has a higher Auslander–Reiten theory. Furthermore, we give a bijection between n-precluster tilting subcategories and n-minimal Auslander–Gorenstein algebras, which is a higher dimensional analog of Auslander–Solberg correspondence [8] as well as a Gorenstein analog of n-Auslander correspondence [22] . The Auslander–Reiten theory associated to an n-precluster tilting subcategory is used to classify the n-minimal Auslander–Gorenstein algebras into four disjoint classes. Our method is based on relative homological algebra due to Auslander–Solberg.

Dolgachev surfaces are simply connected minimal elliptic surfaces with $p_g=q=0$ and of Kodaira dimension 1. These surfaces were constructed by logarithmic transformations of rational elliptic surfaces. In this paper, we explain the construction of Dolgachev surfaces via $\mathbb Q$-Gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities. This construction is based on the paper by Lee-Park. Also, some exceptional bundles on Dolgachev surfaces associated with $\mathbb Q$-Gorenstein smoothing are constructed based on the idea of Hacking. In the case if Dolgachev surfaces were of type $(2,3)$, we describe the Picard group and present an exceptional collection of maximal length. Finally, we prove that the presented exceptional collection is not full, hence there exist a nontrivial phantom category in the derived category., Comment: 35 pages; 3 figures; exposition improved; Adv. Math. (to appear)

An Okounkov body is a convex subset of Euclidean space associated to a divisor on a smooth projective variety with respect to an admissible flag. In this paper, we recover the asymptotic base loci from the Okounkov bodies by studying various asymptotic invariants such as the asymptotic valuations and the moving Seshadri constants. Consequently, we obtain the nefness and ampleness criteria of divisors in terms of the Okounkov bodies. Furthermore, we compute the divisorial Zariski decomposition by the Okounkov bodies, and find upper and lower bounds for moving Seshadri constants given by the size of simplexes contained in the Okounkov bodies., 17 pages. Final version

We give criterions for the existence of toric conical Kahler–Einstein and Kahler–Ricci soliton metrics on any toric manifold in relation to the greatest Ricci and Bakry–Emery–Ricci lower bound. We also show that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with conical Kahler–Einstein metrics in the Gromov–Hausdorff topology.

We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: $\{ (x_1,x_2,x_3): 0, Comment: 53 pages, 5 figures

In this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we show that their category is equivalent to a certain category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantum Toric Varieties. Unlike classical toric varieties, quantum toric varieties admit moduli and we define their moduli spaces, prove that these spaces are orbifolds and, in favorable cases, up to homotopy, they admit a complex structure.

Suppose that R is an analytically irreducible or excellent local domain with maximal ideal m R . We consider multiplicities and mixed multiplicities of R by filtrations of m R -primary ideals. We show that the theorem of Teissier, Rees and Sharp, and Katz, characterizing equality in the Minkowski inequality for multiplicities of ideals, is true for divisorial filtrations, and for the larger category of bounded filtrations. This theorem is not true for arbitrary filtrations of m R -primary ideals.

We prove a Cohen-Macaulay version of a result by Avramov-Golod and Frankild-J{\o}rgensen about Gorenstein rings, showing that if a noetherian ring $A$ is Cohen-Macaulay, and $a_1,\dots,a_n$ is any sequence of elements in $A$, then the Koszul complex $K(A;a_1,\dots,a_n)$ is a Cohen-Macaulay DG-ring. We further generalize this result, showing that it also holds for commutative DG-rings. In the process of proving this, we develop a new technique to study the dimension theory of a noetherian ring $A$, by finding a Cohen-Macaulay DG-ring $B$ such that $\mathrm{H}^0(B) = A$, and using the Cohen-Macaulay structure of $B$ to deduce results about $A$. As application, we prove that if $f:X \to Y$ is a morphism of schemes, where $X$ is Cohen-Macaulay and $Y$ is nonsingular, then the homotopy fiber of $f$ at every point is Cohen-Macaulay. As another application, we generalize the miracle flatness theorem. Generalizations of these applications to derived algebraic geometry are also given., Comment: 26 pages, final version, to appear in Advances in Mathematics

Combinatorial mixed valuations associated to translation-invariant valuations on polytopes are introduced. In contrast to the construction of mixed valuations via polarization, combinatorial mixed valuations reflect and often inherit properties of inhomogeneous valuations. In particular, it is shown that under mild assumptions combinatorial mixed valuations are monotone and hence nonnegative. For combinatorially positive valuations, this has strong computational implications. Applied to the discrete volume, the results generalize and strengthen work of Bihan (2015) on discrete mixed volumes. For rational polytopes, it is proved that combinatorial mixed monotonicity is equivalent to monotonicity. Stronger even, a conjecture is substantiated that combinatorial mixed monotonicity implies the homogeneous monotonicity in the sense of Bernig--Fu (2011)., Comment: 17 pages, minor changes, accepted for publication in Adv. Math

On a smooth variety, Serre's intersection formula computes intersection multiplicities via an alternating sum of the lengths of Tor groups. When the variety is singular, the corresponding sum can be a divergent series. But there are alternate geometric approaches for assigning (often fractional) intersection multiplicities in some singular settings. Our motivating question comes from Fulton, who asks whether an analytic continuation of the divergent series from Serre's formula can be related to these fractional multiplicities. By applying work of Avramov and Buchweitz, we positively answer Fulton's question in the context of graded rings., Comment: 8 pages, 1 figure

Macaulay's Inverse System gives an effective method to construct Artinian Gorenstein k-algebras. To date a general structure for Gorenstein k-algebras of any dimension (and codimension) is not understood. In this paper we extend Macaulay's correspondence characterizing the submodules of the divided power ring in one-to-one correspondence with Gorenstein d-dimensional k-algebras. We discuss effective methods for constructing Gorenstein graded rings. Several examples illustrating our results are given., 19 pages, to appear in Advances in Mathematics

The main aim of this article is to study the relation between F-injective singularity and the Frobenius closure of parameter ideals in Noetherian rings of positive characteristic. The paper consists of the following themes, including many other topics. (1) We prove that if every parameter ideal of a Noetherian local ring of prime characteristic p > 0 is Frobenius closed, then it is F-injective. (2) We prove a necessary and sufficient condition for the injectivity of the Frobenius action on H m i ( R ) for all i ≤ f m ( R ) , where f m ( R ) is the finiteness dimension of R. As applications, we prove the following results. (a) If the ring is F-injective, then every ideal generated by a filter regular sequence, whose length is equal to the finiteness dimension of the ring, is Frobenius closed. It is a generalization of a recent result of Ma and which is stated for generalized Cohen–Macaulay local rings. (b) Let ( R , m , k ) be a generalized Cohen–Macaulay ring of characteristic p > 0 . If the Frobenius action is injective on the local cohomology H m i ( R ) for all i dim ⁡ R , then R is Buchsbaum. This gives an answer to a question of Takagi. (3) We consider the problem when the union of two F-injective closed subschemes of a Noetherian F p -scheme is F-injective. Using this idea, we construct an F-injective local ring R such that R has a parameter ideal that is not Frobenius closed. This result adds a new member to the family of F-singularities. (4) We give the first ideal-theoretic characterization of F-injectivity in terms the Frobenius closure and the limit closure. We also give an answer to the question about when the Frobenius action on the top local cohomology is injective.

In this paper we show Whitney's fibering conjecture in the real and complex, local analytic and global algebraic cases. For a given germ of complex or real analytic set, we show the existence of a stratification satisfying a strong (real arc-analytic with respect to all variables and analytic with respect to the parameter space) trivialization property along each stratum. We call such a trivialization arc-wise analytic and we show that it can be constructed under the classical Zariski algebro-geometric equisingularity assumptions. Using a slightly stronger version of the Zariski equisingularity, we show the existence of Whitney's stratified fibration, satisfying the conditions (b) of Whitney and (w) of Verdier. Our construction is based on the Puiseux with parameter theorem and a generalization of Whitney's interpolation. For algebraic sets our construction gives a global stratification. We also present several applications of the arc-wise analytic trivialization, mainly to the stratification theory and the equisingularity of analytic set and function germs. In the real algebraic case, for an algebraic family of projective varieties, we show that the Zariski equisingularity implies local constancy of the associated weight filtration.

We prove that if ϕ : ( X , 0 ) → ( X , 0 ) is a finite endomorphism of an isolated singularity such that deg ⁡ ( ϕ ) ≥ 2 and ϕ is etale in codimension 1, then X is Q -Gorenstein and log canonical.

We determine the Bernstein–Sato polynomials for the ideal of maximal minors of a generic m × n matrix, as well as for that of sub-maximal Pfaffians of a generic skew-symmetric matrix of odd size. As a corollary, we obtain that the Strong Monodromy Conjecture holds in these two cases.

We define the Frobenius limit of a module over a ring of prime characteristic to be the limit of the normalized Frobenius direct images in a certain Grothendieck group. When a finite group acts on a polynomial ring, we calculate this limit for all the modules over the twisted group algebra that are free over the polynomial ring; we also calculate the Frobenius limit for the restriction of these to the ring of invariants. As an application, we generalize the description of the generalized $F$-signature of a ring of invariants by the second author and Nakajima to the modular case., 25 pages

We show for a ring R of weak global dimension at most one that there is a bijection between the smashing subcategories of its derived category and the equivalence classes of homological epimorphisms starting in R. If, moreover, R is commutative, we prove that the compactly generated localizing subcategories correspond precisely to flat epimorphisms. We also classify smashing localizations of the derived category of any valuation domain, and provide an easy criterion for the Telescope Conjecture (TC) for any commutative ring of weak global dimension at most one. As a consequence, we show that the TC holds for any commutative von Neumann regular ring R, and it holds precisely for those Pr\"ufer domains which are strongly discrete., Comment: 45 pages; version 2: several changes in the presentation (a section on the homotopy category of dg algebras became an appendix, more explanation added at various places of the text), main results unchanged

In this paper, we study extensions of graded affine Hecke algebra modules. In particular, based on an explicit projective resolution on graded affine Hecke algebra modules, we prove a duality result for Ext-groups. This duality result with an Ind-Res resolution gives an algebraic proof of the fact that all higher Ext-groups between discrete series vanish., Comment: 36 pages, v2: Some results and proofs are improved. Sections 5,6,7 in v2 are new. Section 5 in v1 is completely removed and may appear elsewhere; v3: close to published version

We describe a general correspondence between injective (resp. projective) recollements of triangulated categories and injective (resp. projective) cotorsion pairs. This provides a model category description of these recollement situations. Our applications focus on displaying several recollements that glue together various full subcategories of K(R), the homotopy category of chain complexes of modules over a general ring R. When R is (left) Noetherian ring, these recollements involve complexes built from the Gorenstein injective modules. When R is a (left) coherent ring for which all flat modules have finite projective dimension we obtain the duals. These results extend to a general ring R by replacing the Gorenstein modules with the Gorenstein AC-modules introduced recently in the work of Bravo-Gillespie-Hovey. We also see that in any abelian category with enough injectives, the Gorenstein injective objects enjoy a maximality property in that they contain every other class making up the right half of an injective cotorsion pair., 45 pages. Revisions from previous version include: 0) A major rewrite resulting in a completely different numbering of Lemmas, Propositions, etc. 1) removal of a false claim in Example 4.8 on extending the recollement of Krause. 2) A converse to the main theorem. 3) An extension of the Gorenstein applications to arbitrary rings using the Gorenstein AC-modules

In this paper, we give a new version of the modified Futaki invariant for a test configuration associated to the soliton action on a Fano manifold. Our version will naturally come from toric test configurations defined by Donaldson for toric manifolds. As an application, we show that the modified K-energy is proper for toric invariant Kahler potentials on a toric Fano manifold.

In this paper, we show that the secant variety to a smooth projective variety embedded by a sufficiently positive line bundle is normal. As an application, we deduce that the secant variety to a general canonical curve of genus at least 7 is normal., Comment: 14 pages; To appear in Advances in Mathematics; Lemma added in section 2 to improve readability

We explore the classical Lech's inequality relating the Hilbert--Samuel multiplicity and colength of an $\mathfrak{m}$-primary ideal in a Noetherian local ring $(R,\mathfrak{m})$. We prove optimal versions of Lech's inequality for sufficiently deep ideals in characteristic $p>0$, and we conjecture that they hold in all characteristics. Our main technical result shows that if $(R,\mathfrak{m})$ has characteristic $p>0$ and $\widehat{R}$ is reduced, equidimensional, and has an isolated singularity, then for any sufficiently deep $\mathfrak{m}$-primary ideal $I$, the colength and Hilbert--Kunz multiplicity of $I$ are sufficiently close to each other. More precisely, for all $\varepsilon>0$, there exists $N\gg0$ such that for any $I\subseteq R$ with $l(R/I)>N$, we have $(1-\varepsilon)l(R/I)\leq e_{HK}(I)\leq(1+\varepsilon)l(R/I)$., 28 pages, final version

We discuss finitely graded Iwanaga-Gorenstein (IG) algebras $A$ and representation theory of their (graded) Cohen-Macaulay (CM) modules. By quasi-Veronese algebra construction, in principle, we may reduce our study to the case where $A$ is a trivial extension algebra $A = \Lambda \oplus C$ with the grading $ \text{deg} \Lambda = 0, \ \text{deg} C = 1$. In the previous study, we gave a necessary and sufficient condition that $A$ is IG in terms of $\Lambda$ and $C$ by using derived tensor products and derived Homs. For simplicity, we assume that $\Lambda$ is of finite global dimension in the sequel. In this paper, we show that the condition that $A$ is IG, has a triangulated categorical interpretation. We prove that if $A$ is IG, then the graded stable category $\underline{\mathsf{CM}}^{\Bbb{Z}} A$ of CM-modules is realized as an admissible subcategory of the derived category $\mathsf{D}^{\mathrm{b}}(\text{mod } \Lambda)$. As a corollary, we deduce that the Grothendieck group $K_{0}(\underline{\mathsf{CM}}^{\Bbb{Z}} A)$ is free of finite rank. We give several applications. Among other things, for a path algebra $\Lambda= \Bbb{k} Q$ of an $A_{2}$ or $A_{3}$ quiver Q, we give a complete list of $\Lambda$-$\Lambda$-bimodule $C$ such that $\Lambda \oplus C$ is IG (resp. of finite global dimension) by using the triangulated categorical interpretation mentioned above., Comment: 43 pages. Minor revisions

This study examines the finite $F$-representation type (abbr. FFRT) property of a two-dimensional normal graded ring $R$ in characteristic $p>0$, using notions from the theory of algebraic stacks. Given a graded ring $R$, we consider an orbifold curve $\mathfrak C$, which is a root stack over the smooth curve $C=\text{Proj} R$, such that $R$ is the section ring associated with a line bundle $L$ on $\mathfrak C$. The FFRT property of $R$ is then rephrased with respect to the Frobenius push-forwards $F^e_*(L^i)$ on the orbifold curve $\mathfrak C$. As a result, we see that if the singularity of $R$ is not log terminal, then $R$ has FFRT only in exceptional cases where the characteristic $p$ divides a weight of $\mathfrak C$., 25 pages, exposition on stacks added, to appear in Adv. Math

A special case of the Menshov--Rademacher theorem implies for almost all polynomials $x_1Z+\ldots +x_d Z^{d} \in {\mathbb R}[Z]$ of degree $d$ for the Weyl sums satisfy the upper bound $$ \left| \sum_{n=1}^{N}\exp\left(2\pi i \left(x_1 n+\ldots +x_d n^{d}\right)\right) \right| \leqslant N^{1/2+o(1)}, \qquad N\to \infty. $$ Here we investigate the exceptional sets of coefficients $(x_1, \ldots, x_d)$ with large values of Weyl sums for infinitely many $N$, and show that in terms of the Baire categories and Hausdorff dimension they are quite massive, in particular of positive Hausdorff dimension in any fixed cube inside of $[0,1]^d$. We also use a different technique to give similar results for sums with just one monomial $xn^d$. We apply these results to show that the set of poorly distributed modulo one polynomials is rather massive as well., Comment: 44 pages, 2 figures

In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is to determine the minimal models within any birational class. In this paper we show that the generic noncommutative projective plane (corresponding to the three dimensional Sklyanin algebra R) as well as noncommutative analogues of P^1 x P^1 and of the Hirzebruch surface F_2 (arising from Van den Bergh's quadrics R) satisfy very strong minimality conditions. Translated into an algebraic question, where one is interested in a maximality condition, we prove the following theorem. Let R be a Sklyanin algebra or a Van den Bergh quadric that is infinite dimensional over its centre and let A be any connected graded noetherian maximal order containing R, with the same graded quotient ring as R. Then, up to taking Veronese rings, A is isomorphic to R. Secondly, let T be an elliptic algebra (that is, the coordinate ring of a noncommutative surface containing an elliptic curve). Then, under an appropriate homological condition, we prove that every connected graded noetherian overring of T is obtained by blowing down finitely many lines (line modules)., Comment: 42 pages; Third and final version to appear in Advances in Mathematics: minor revisions from second version

We give an intrinsic definition of toric symplectic stacks, and show that they are classified by simple convex polytopes equipped with some additional combinatorial data. This generalizes Delzant's classification of toric symplectic manifolds. As an application, we show that any toric symplectic stack can be deformed to an ineffective toric orbifold., 35 pages, 2 figures

Let $X \subseteq \mathbb{P}^r$ be a non-degenerate smooth projective variety of dimension $n$, codimension $e$, and degree $d$ defined over an algebraically closed field of characteristic zero. In this paper, we first show that $\text{reg} (\mathcal{O}_X) \leq d-e$, and classify the extremal and the next to extremal cases. Our result reduces the Eisenbud-Goto regularity conjecture for the smooth case to the problem finding a Castelnuovo-type bound for normality. It is worth noting that McCullough-Peeva recently constructed counterexamples to the regularity conjecture by showing that $\text{reg} (\mathcal{O}_X)$ is not even bounded above by any polynomial function of $d$ when $X$ is not smooth. For a normality bound in the smooth case, we establish that $\text{reg}(X) \leq n(d-2)+1$, which improves previous results obtained by Mumford, Bertram-Ein-Lazarsfeld, and Noma. Finally, by generalizing Mumford's method on double point divisors, we prove that $\text{reg}(X) \leq d-1+m$, where $m$ is an invariant arising from double point divisors associated to outer general projections. Using double point divisors associated to inner projection, we also obtain a slightly better bound for $\text{reg}(X)$ under suitable assumptions., Comment: 23 pages. This paper has been largely rewritten after McCullough-Peeva's counterexamples to the Eisenbud-Goto regularity conjecture, which appeared in J. Amer. Math. Soc. in 2018. We also added new results on the regularity of smooth projective varieties of arbitrary dimension

Suppose that $R$ is an excellent local domain with maximal ideal $m_R$. The theory of multiplicities and mixed multiplicities of $m_R$-primary ideals extends to (possibly non Noetherian) filtrations of $R$ by $m_R$-primary ideals, and many of the classical theorems for $m_R$-primary ideals continue to hold for filtrations. The celebrated theorems involving inequalities continue to hold for filtrations, but the good conclusions that hold in the case of equality for $m_R$-primary ideals do not hold for filtrations. In this article, we consider multiplicities and mixed multiplicities of $R$ by $m_R$-primary divisorial filtrations. We show that some important theorems on equalities of multiplicities and mixed multiplicities of $m_R$-primary ideals, which are not true in general for filtrations, are true for divisorial filtrations. We prove that a theorem of Rees showing that if there is an inclusion of $m_R$-primary ideals $I\subset I'$ with the same multiplicity then $I$ and $I'$ have the same integral closure also holds for divisorial filtrations. This theorem does not hold for arbitrary filtrations. We show that the Teissier Rees Sharp Katz theorem on equality in the Minkowski inequality holds for divisorial filtrations in an excellent domain of dimension two. We also show that the mixed multiplicities of divisorial filtrations are anti-positive intersection products on a suitable normal scheme $X$ birationally dominating $R$, when $R$ is an algebraic local domain., 36 pages

On the product elliptic threefold $X = C \times S$ where $C$ is an elliptic curve and $S$ is a K3 surface of Picard rank 1, we define a notion of limit tilt stability, which satisfies the Harder-Narasimhan property. We show that under the Fourier-Mukai transform $\Phi$ on $D^b(X)$ induced by the classical Fourier-Mukai transform on $D^b(C)$, a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is taken to a limit tilt stable object. We also show that a limit tilt semistable object on $X$ is taken by $\Phi$ to a slope semistable sheaf, up to modification by the transform of a codimension 2 sheaf., Comment: 39 pages. Minor typos corrected. To appear in Adv. Math

In this article, we characterize plurisubharmonic functions of Lelong number one at the origin, such that the germ of the associated multiplier ideal sheaf is nontrivial: in arbitrary complex dimension, their singularity must be the sum of a germ of smooth divisor and of a plurisubharmonic function with zero Lelong number. We also present a new proof of the related well known integrability criterion due to Skoda., Comment: 14 pages, 0 figures. Revised version

Let ( R , m ) be a local Noetherian ring, let M be a finitely generated R-module and let ( F • , ∂ • ) be a free resolution of M. We find a uniform bound h such that the Artin–Rees containment I n F i ∩ Im ∂ i + 1 ⊆ I n − h Im ∂ i + 1 holds for all integers i ≥ d , for all integers n ≥ h , and for all ideals I of R. In fact, we show that a considerably stronger statement holds. The uniform bound h holds for all ideals and all resolutions of dth syzygy modules. In order to prove our statements, we introduce the concept of Koszul annihilating sequences.

The (4k+2)-dimensional Kervaire manifold is a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product of two (2k+1)-dimensional spheres. We show that a finite group of odd order acts freely on a Kervaire manifold if and only if it acts freely on the corresponding product of spheres. If the Kervaire manifold M is smoothable, then each smooth structure on M admits a free smooth involution. If k + 1 is not a 2-power, then the Kervaire manifold in dimension 4k+2 does not admit any free TOP involutions. Free "exotic" (PL) involutions are constructed on the Kervaire manifolds of dimensions 30, 62, and 126. Each smooth structure on the 30-dimensional Kervaire manifold admits a free Z/2 x Z/2 action., 36 pages. Final version: Advances in Mathematics (to appear)

Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of PDEs. We explain these conjectures and verify them in some cases.

Consider a graph C ⁎ -algebra C ⁎ ( E ) with a purely infinite ideal I (possibly all of C ⁎ ( E ) ) such that I has only finitely many ideals and C ⁎ ( E ) / I is approximately finite dimensional. We prove that the nuclear dimension of C ⁎ ( E ) is 1. If I has infinitely many ideals, then the nuclear dimension of C ⁎ ( E ) is either 1 or 2.

We define a derived version of Mazur's Galois deformation ring. It is a pro-simplicial ring $\mathcal{R}$ classifying deformations of a fixed Galois representation to simplicial coefficient rings; its zeroth homotopy group $\pi_0 \mathcal{R}$ recovers Mazur's deformation ring. We give evidence that these rings $\mathcal{R}$ occur in the wild: For suitable Galois representations, the Langlands program predicts that $\pi_0 \mathcal{R}$ should act on the homology of an arithmetic group. We explain how the Taylor--Wiles method can be used to upgrade such an action to a graded action of $\pi_* \mathcal{R}$ on the homology., Comment: This is the submitted version. A few minor changes from version 1. Metadata and added acknowledgement in v3