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We introduce a new class of commutative noetherian DG-rings which generalizes the class of regular local rings. These are defined to be local DG-rings (A , m ¯) such that the maximal ideal m ¯ ⊆ H 0 (A) can be generated by an A -regular sequence. We call these DG-rings sequence-regular DG-rings, and make a detailed study of them. Using methods of Cohen-Macaulay differential graded algebra, we prove that the Auslander-Buchsbaum-Serre theorem about localization generalizes to this setting. This allows us to define global sequence-regular DG-rings, and to introduce this regularity condition to derived algebraic geometry. It is shown that these DG-rings share many properties of classical regular local rings, and in particular we are able to construct canonical residue DG-fields in this context. Finally, we show that sequence-regular DG-rings are ubiquitous, and in particular, any eventually coconnective derived algebraic variety over a perfect field is generically sequence-regular. [ABSTRACT FROM AUTHOR]

The theory of extensive categories determines in particular the notion of separable MV-algebra (equivalently, of separable unital lattice-ordered Abelian group). We establish the following structure theorem: An MV-algebra is separable if, and only if, it is a finite product of algebras of rational numbers—i.e., of subalgebras of the MV-algebra [ 0 , 1 ] ∩ Q. Beyond its intrinsic algebraic interest, this research is motivated by the long-term programme of developing the algebraic geometry of the opposite of the category of MV-algebras, in analogy with the classical case of commutative K -algebras over a field K. [ABSTRACT FROM AUTHOR]

Since the 1974 paper by Peskine and Szpiro, liaison theory via complete intersections, and more generally via Gorenstein varieties, has become a standard tool kit in commutative algebra and algebraic geometry, allowing to compare algebraic features of linked varieties. In this paper we develop a liaison theory via quasi-Gorenstein varieties, a much broader class than Gorenstein varieties. As applications, we derive a connectedness property of quasi-Gorenstein subspace arrangements generalizing previous results by Benedetti and the first author, and we deduce the classical topological Lefschetz duality via the Stanley-Reisner correspondence. [ABSTRACT FROM AUTHOR]

Varieties of minimal degree and del Pezzo varieties are basic objects in projective algebraic geometry. Those varieties have been characterized and classified for a long time in many aspects. Motivated by the question "which varieties are the most basic and simplest except the above two kinds of varieties in view of geometry and syzygies?", we give an upper bound of the graded Betti numbers in the quadratic strand and characterize the extremal cases. The extremal varieties of dimension n , codimension e , and degree d are exactly characterized by the following two types: (i) Varieties with d = e + 2 , depth X = n , and Green-Lazarsfeld index a (X) = 0 , (ii) Arithmetically Cohen-Macaulay varieties with d = e + 3. This is a generalization of G. Castelnuovo, G. Fano, and E. Park's results on the number of quadrics and an extension of the characterizations of varieties of minimal degree and del Pezzo varieties in view of linear syzygies of quadrics due to K. Han and S. Kwak ([6,8,30,16]). In addition, we show that every variety X that belongs to (i) or (ii) is always contained in a unique rational normal scroll Y as a divisor. Also, we describe the divisor class of X in Y. [ABSTRACT FROM AUTHOR]

In this paper, we use algebro-geometric methods in order to derive classification results for so-called D -bialgebra structures on the power series algebra A 〚 z 〛 for certain central simple non-associative algebras A. These structures are closely related to a version of the classical Yang-Baxter equation (CYBE) over A. If A is a Lie algebra, we obtain new proofs for pivotal steps in the known classification of non-degenerate topological Lie bialgebra structures on A 〚 z 〛 as well as of non-degenerate solutions of the usual CYBE. If A is associative, we achieve the classification of non-triangular topological balanced infinitesimal bialgebra structures on A 〚 z 〛 as well as of all non-degenerate solutions of an associative version of the CYBE. [ABSTRACT FROM AUTHOR]

In this paper we construct a tropical moduli space parametrizing roots of divisors on tropical curves. We study the relation between this space and the skeleton of Jarvis moduli space of nets of limit roots on stable curves. We show that the combinatorics of the moduli space of tropical roots is governed by the poset of flows, a poset parametrizing certain flows on graphs. [ABSTRACT FROM AUTHOR]

In this work, we study the Kähler-Ricci flow on rational homogeneous varieties exploring the interplay between projective algebraic geometry and representation theory which underlies the classical Borel-Weil theorem. By using elements of representation theory of semisimple Lie groups and Lie algebras, we give an explicit description for all solutions of the Kähler-Ricci flow with homogeneous initial condition. This description enables us to compute explicitly the maximal existence time for any solution starting at a homogeneous Kähler metric and obtain explicit upper and lower bounds for several geometric quantities along the flow, including curvatures, volume, diameter, and the first non-zero eigenvalue of the Laplacian. As an application of our main result, we investigate the relationship between certain numerical invariants associated with ample divisors and numerical invariants arising from solutions of the Kähler-Ricci flow in the homogeneous setting. [ABSTRACT FROM AUTHOR]

This paper gives a new, simplified presentation of the classical pure braid group. The generators are given by the squares of the longest elements over connected subgraphs, and we prove that the only relations are either commutators or certain palindromic length 5 box relations. This presentation is motivated by twist functors in algebraic geometry, but the proof is entirely Coxeter-theoretic. We also prove that the analogous set does not generate for all Coxeter arrangements, which in particular answers a question of Donovan and Wemyss. [ABSTRACT FROM AUTHOR]

We study the complexity of multiplication of two elements in a finite field extension given by their coordinates in a normal basis. We show how to control this complexity using the arithmetic and geometry of algebraic curves. [ABSTRACT FROM AUTHOR]

In noncommutative algebraic geometry, noncommutative quadric hypersurfaces are major objects of study. In this paper, we focus on studying noncommutative conics Pro j nc A embedded into Calabi-Yau quantum projective planes. In particular, we give complete classifications of homogeneous coordinate algebras A of noncommutative conics up to isomorphism of graded algebras, and of noncommutive conics Pro j nc A up to isomorphism of noncommutative schemes. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field. Building upon previous work, we classify, up to isomorphism of algebras, quadratic graded twisted tensor products of K [ x , y ] and K [ z ]. When such an algebra is Artin-Schelter regular, we identify its point scheme and type, in the sense of [6]. We also describe which three-dimensional Sklyanin algebras contain a subalgebra isomorphic to a quantum P 1 , and we show that every algebra in this family is a graded twisted tensor product of K − 1 [ x , y ] and K [ z ]. [ABSTRACT FROM AUTHOR]

Real Nullstellensatz is a classical result from Real Algebraic Geometry. It has recently been extended to quaternionic polynomials by Alon and Paran [1]. The aim of this paper is to extend their Quaternionic Nullstellensatz to matrix polynomials. We also obtain an improvement of the Real Nullstellensatz for matrix polynomials from [4] in the sense that we simplify the definition of a real left ideal. We use the methods from the proof of the matrix version of Hilbert's Nullstellensatz [5] and we obtain their extensions to a mildly non-commutative case and to the real case. [ABSTRACT FROM AUTHOR]

We introduce and describe the 2-category Grt ♭ of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories ⊠ restricts nicely to Grt ♭. Then, we characterize exponentiable objects with respect to ⊠: these are the continuous Grothendieck categories. In particular, locally finitely presentable Grothendieck categories are exponentiable. Consequently, we have that, for a quasi-compact quasi-separated scheme X , the category of quasi-coherent sheaves Qcoh (X) is exponentiable. Finally, we provide a family of examples and concrete computations of exponentials. [ABSTRACT FROM AUTHOR]

Wall-crossing phenomena are ubiquitous in many problems of algebraic geometry and theoretical physics. Various ways to encode the relevant information and the need to track the changes under the variation of parameters lead to rather complicated transformation rules and non-trivial combinatorial problems. In this paper we propose a framework, reminiscent of collections and plethysms in the theory of operads, that conceptualizes those transformation rules. As an application we obtain new streamlined proofs of some existing wall-crossing formulas as well as some new formulas related to attractor invariants. [ABSTRACT FROM AUTHOR]

We show that there is an equivalence of categories between Lie-Rinehart algebras over a commutative algebra O and homotopy equivalence classes of negatively graded Lie ∞-algebroids over their resolutions (=acyclic Lie ∞-algebroids). This extends to a purely algebraic setting the construction of the universal Q -manifold of a locally real analytic singular foliation of [26,28]. In particular, it makes sense for the universal Lie ∞-algebroid of every singular foliation, without any additional assumption, and for Androulidakis-Zambon singular Lie algebroids. Also, to any ideal I ⊂ O preserved by the anchor map of a Lie-Rinehart algebra A , we associate a homotopy equivalence class of negatively graded Lie ∞-algebroids over complexes computing Tor O (A , O / I). Several explicit examples are given. [ABSTRACT FROM AUTHOR]

Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of noncommutative matrix factorization for an arbitrary nonzero non-unit element of a ring. First we show that the category of noncommutative graded matrix factorizations is invariant under the operation called twist (this result is a generalization of the result by Cassidy-Conner-Kirkman-Moore). Then we give two category equivalences involving noncommutative matrix factorizations and totally reflexive modules (this result is analogous to the famous result by Eisenbud for commutative hypersurfaces). As an application, we describe indecomposable noncommutative graded matrix factorizations over skew exterior algebras. [ABSTRACT FROM AUTHOR]

A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr (x j). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials. [ABSTRACT FROM AUTHOR]

In this note we discuss three notions of dimension for triangulated categories: Rouquier dimension, diagonal dimension and Serre dimension. We prove some basic properties of these dimensions, compare them and discuss open problems. [ABSTRACT FROM AUTHOR]

More than four decades ago, Eisenbud, Khimšiašvili, and Levine introduced an analogue in the algebro-geometric setting of the notion of local degree from differential topology. Their notion of degree, which we call the EKL-degree, can be thought of as a refinement of the usual notion of local degree in algebraic geometry that works over non-algebraically closed base fields, taking values in the Grothendieck-Witt ring. In this note, we compute the EKL-degree at the origin of certain finite covers f : A n → A n induced by quotients under actions of Weyl groups. We use knowledge of the cohomology ring of partial flag varieties as a key input in our proofs, and our computations give interesting explicit examples in the field of A 1 -enumerative geometry. [ABSTRACT FROM AUTHOR]

We prove the existence theorem for basic elements in the quasi-projective case, extending results of Eisenbud-Evans and Bruns from the affine case. We give several geometric applications. For example, we show that every local complete intersection of pure codimension two in a smooth projective variety of dimension d over an infinite field is the degeneracy locus of d − 1 sections of a rank d bundle. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show that for a complete intersection curve C in projective space (other than a few exceptions stated below), any morphism f : C → P r satisfying deg ⁡ f ⁎ O P r (1) < deg ⁡ C is obtained by projection from a linear space. In particular, we obtain bounds on the gonality of such curves and compute the gonality of general complete intersection curves. We also prove a special case of one of the well-known Cayley-Bacharach conjectures posed by Eisenbud, Green, and Harris. [ABSTRACT FROM AUTHOR]

Let C ⊂ P r be a general curve of genus g embedded via a general linear series of degree d. The Maximal Rank Conjecture asserts that the restriction maps H 0 (O P r (m)) → H 0 (O C (m)) are of maximal rank; this determines the Hilbert function of C. In this paper, we prove an analogous statement for the union of hyperplane sections of general curves. More specifically, if H ⊂ P r is a general hyperplane, and C 1 , C 2 , ... , C n are general curves, we show H 0 (O H (m)) → H 0 (O (C 1 ∪ C 2 ∪ ⋯ ∪ C n) ∩ H (m)) is of maximal rank, except for some counterexamples when m = 2. As explained in [5] , this result plays a key role in the author's proof of the Maximal Rank Conjecture [7]. [ABSTRACT FROM AUTHOR]

Given a quasi-compact and quasi-separated (qcqs) scheme X of finite type over a Noetherian ring R having an ample family of line bundles, we construct a closed embedding of X into a smooth (qcqs) scheme over R having an ample family of line bundles. Such a smooth scheme arises as an open subscheme of the multihomogeneous Proj of a Z n -graded ring. [ABSTRACT FROM AUTHOR]

Let L be a tensor product of two very ample line bundles on the smooth projective complex threefold X. Under the hypothesis that H 0 (X , K X (L)) ≠ 0 , we show that the restriction map r : Pic X → Pic Y is an isomorphism for very general Y in the linear system | L |. For such L this result recovers the Noether-Lefschetz theorem of Moishezon, who extended the original topological and Hodge-theoretic arguments of Lefschetz. We give a degeneration argument which is almost completely algebraic in nature. [ABSTRACT FROM AUTHOR]

Let V = C 2 ⊗ C 2 ⊗ C 2 be the space of 2 × 2 × 2 hypermatrices, endowed with the natural group action of GL = GL 2 (C) × GL 2 (C) × GL 2 (C). The category of GL-equivariant coherent left D -modules on V is equivalent to the category of representations of a quiver with relations. In this article, we give a construction of each simple object and study their GL-equivariant structure. Using this information, we go on to explicitly describe the corresponding quiver with relations. As an application, we compute all iterations of local cohomology with support in the orbit closures of V. [ABSTRACT FROM AUTHOR]

In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorem, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. These improve known such results. [ABSTRACT FROM AUTHOR]

Two classical results in algebraic geometry are that the branch curve of a del Pezzo surface of degree 1 can be embedded as a space sextic curve in P 3 and that every space sextic curve has exactly 120 tritangents corresponding to its odd theta characteristics. In this paper we revisit both results from the computational perspective. Specifically, we give an algorithm to construct space sextic curves that arise from blowing up P 2 at eight points and provide algorithms to compute the 120 tritangents and their Steiner system of any space sextic. Furthermore, we develop efficient inverses to the aforementioned methods. We present an algorithm to either reconstruct the original eight points in P 2 from a space sextic or certify that this is not possible. Moreover, we extend a construction of Lehavi [8] which recovers a space sextic from its tritangents and Steiner system. All algorithms in this paper have been implemented in magma. [ABSTRACT FROM AUTHOR]

Let D be a division ring, n ≥ 3 an integer, and P n (D) the poset of all n × n idempotent matrices over D with the partial order defined by P ≤ Q if P Q = Q P = P. Let T ∈ M n (D) be an invertible matrix and σ : D → D an endomorphism (τ : D → D an anti-endomorphism). For any P ∈ P n (D) we denote by P σ (P τ) the idempotent matrix obtained from P by applying σ (τ) entrywise. The map ϕ : P n (D) → P n (D) defined by ϕ (P) = T P σ T − 1 , P ∈ P n (D) (ϕ (P) = T t (P τ) T − 1 , P ∈ P n (D)) is an injective map preserving order in both directions. Every such map is called a standard map. It has been known before that if D is an EAS division ring, then every injective order preserving map ϕ : P n (D) → P n (D) is either standard or of a very special degenerate form. In this paper we use some ideas from geometry of algebraic homogeneous spaces and elementary field theory to give examples showing that the EAS assumption is indispensable. Then we define generalized standard maps and using them we describe the general form of injective order preservers on P n (D) for an arbitrary division ring D. Our proof is shorter than the original one for the special case of EAS division rings. Under somewhat stronger assumptions we get two characterizations of standard maps. All the results are optimal as shown by counterexamples. [ABSTRACT FROM AUTHOR]

We address the problem to determine the flat limit of the collision of fat points in P n. We give a description of the limit scheme in many cases, in particular in low dimension and multiplicities. The problem turns out to be closely related with interpolation theory, and as an application we exploit collisions to prove new cases of Laface-Ugaglia Conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we will give an algebraic proof for determining the sections for the universal pointed hyperelliptic curves C H g , n / k → H g , n / k , when g ≥ 3 and the image of the ℓ -adic cyclotomic character G k → Z × is infinite. Furthermore, we will study the nonabelian phenomena associated to the universal hyperelliptic curves. For example, we will show that the section conjecture holds for C H g / k → H g / k and the unipotent analogue of the conjecture holds for the pointed cases. This work is an extension of Hain's original work on the rational points of universal curves to the hyperelliptic case. [ABSTRACT FROM AUTHOR]

Abstract In the present paper we provide a description of complete Calabi-Yau metrics on the canonical bundle of generalized complex flag manifolds. By means of Lie theory we give an explicit description of complete Ricci-flat Kähler metrics obtained through the Calabi Ansatz technique. We use this approach to provide several explicit examples of noncompact complete Calabi-Yau manifolds, these examples include canonical bundles of non-toric flag manifolds (e.g. Grassmann manifolds and full flag manifolds). [ABSTRACT FROM AUTHOR]

Abstract We present a notion of a random toric surface modeled on a notion of a random graph. We then study some threshold phenomena related to the smoothness of the resulting surfaces. [ABSTRACT FROM AUTHOR]

Abstract Let A be a d × n integer matrix. Gel′fand et al. proved that most A -hypergeometric systems have an interpretation as a Fourier–Laplace transform of a direct image. The set of parameters for which this happens was later identified by Schulze and Walther as the set of not strongly resonant parameters of A. A similar statement relating A -hypergeometric systems to exceptional direct images was proved by Reichelt. In this article, we consider a hybrid approach involving neighborhoods U of the torus of A and consider compositions of direct and exceptional direct images. Our main results characterize for which parameters the associated A -hypergeometric system is the inverse Fourier–Laplace transform of such a "mixed Gauss–Manin" system. In order to describe which U work for such a parameter, we introduce the notions of fiber support and cofiber support of a D -module. If the semigroup ring C [ N A ] is normal, we show that every A -hypergeometric system is "mixed Gauss–Manin". We also give an explicit description of the neighborhoods U which work for each parameter in terms of primitive integral support functions. [ABSTRACT FROM AUTHOR]

Abstract Roughly ten years ago, the following "Gorenstein Interval Conjecture" (GIC) was proposed: Whenever (1 , h 1 , ... , h i , ... , h e − i , ... , h e − 1 , 1) and (1 , h 1 , ... , h i + α , ... , h e − i + α , ... , h e − 1 , 1) are both Gorenstein Hilbert functions for some α ≥ 2 , then (1 , h 1 , ... , h i + β , ... , h e − i + β , ... , h e − 1 , 1) is also Gorenstein, for all β = 1 , 2 , ... , α − 1. Since an explicit characterization of which Hilbert functions are Gorenstein is widely believed to be hopeless, the GIC, if true, would at least provide the existence of a strong, and very natural, structural property for such basic functions in commutative algebra. Before now, very little progress was made on the GIC. The main goal of this note is to prove the case e ≤ 5 , in arbitrary codimension. Our arguments will be in part constructive, and will combine several different tools of commutative algebra and classical algebraic geometry. [ABSTRACT FROM AUTHOR]

Abstract We study the shape of the F-signature function of a d -dimensional quotient singularity k 〚 x 1 , ... , x d 〛 G , and we show that it is a quasi-polynomial. We prove that the second coefficient is always zero and we describe the other coefficients in terms of invariants of the finite acting group G ⊆ Gl (d , k). When G is cyclic, we obtain more specific formulas for the coefficients of the quasi-polynomial, which allow us to compute the general form of the function in several examples. [ABSTRACT FROM AUTHOR]

Any virtually free group H containing no non-trivial finite normal subgroup (e.g., the infinite dihedral group) is a retract of any finitely generated group containing H as a verbally closed subgroup. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field of characteristic zero, and V a hypersurface defined by an irreducible polynomial f with coefficients in K . In this article we prove that an Embedded Deformation of V which admits a Simultaneous Embedded Resolution induces, under certain conditions, a deformation of the space of m -jets V m , m ≥ 0 . An example of an Embedded Deformation of V which admits a Simultaneous Embedded Resolution is a Γ ( f ) -deformation of V , where V has at most one isolated singularity, and f is non-degenerate with respect to the Newton Boundary Γ ( f ) . [ABSTRACT FROM AUTHOR]

Let Z ⊂ A k be an affine scheme over C and J Z its jet scheme. It is well-known that C [ J Z ] , the coordinate ring of J Z , has the structure of a commutative vertex algebra. This paper develops the orbifold theory for C [ J Z ] . A finite-order linear automorphism g of Z acts by vertex algebra automorphisms on C [ J Z ] . We show that C [ J g Z ] , where J g Z is the scheme of g -twisted jets has the structure of a g -twisted C [ J Z ] module. We consider spaces of orbifold coinvariants valued in the modules C [ J g Z ] on orbicurves [ Y / G ] , with Y a smooth projective curve and G a finite group, and show that these are isomorphic to C [ Z G ] . [ABSTRACT FROM AUTHOR]

We introduce a class of strongly étale difference algebras , whose role in the study of difference equations is analogous to the role of étale algebras in the study of algebraic equations. We deduce an improved version of Babbitt's decomposition theorem and we present applications to difference algebraic groups and the compatibility problem. [ABSTRACT FROM AUTHOR]

We prove that there is no non-trivial autoequvivalence of the category of schemes of finite type over Q . [ABSTRACT FROM AUTHOR]

Let R = S / I be a quotient of a standard graded polynomial ring S by an ideal I generated by quadrics. If R is Koszul, a question of Avramov, Conca, and Iyengar asks whether the Betti numbers of R over S can be bounded above by binomial coefficients on the minimal number of generators of I . Motivated by previous results for Koszul algebras defined by three quadrics, we give a complete classification of the structure of Koszul almost complete intersections and, in the process, give an affirmative answer to the above question for all such rings. [ABSTRACT FROM AUTHOR]

In algebraic geometry, Bertini theorems are an extremely important tool. A generalization of the classical theorem to multiplier ideals show that multiplier ideals restrict to a general hyperplane section. The test ideal can be seen to be the characteristic p > 0 analog of the multiplier ideal. However, in this paper it is shown that the same type of Bertini type theorem does not hold for test ideals. [ABSTRACT FROM AUTHOR]

The notion of p g -ideals for normal surface singularities has been proved to be very useful. On the other hand, the core of ideals has been proved to be very important concept and also very mysterious one. However, the computation of the core of an ideal seems to be given only for very special cases. In this paper, we will give an explicit description of the core of p g -ideals of normal surface singularities. As a consequence, we give a characterization of rational singularities using the inclusion of the core of integrally closed ideals. [ABSTRACT FROM AUTHOR]

We investigate torsion exceptional sheaves on a weak del Pezzo surface of degree greater than two whose anticanonical model has at most A n -singularities. We show that every torsion exceptional sheaf can be obtained from a line bundle on a ( − 1 ) -curve by spherical twists. [ABSTRACT FROM AUTHOR]

We investigate the extent to which the Bump–Hoffstein conjecture could be generalized for central coverings of the general linear groups. We provide evidence for such generalized Bump–Hoffstein conjecture by proving some special cases. [ABSTRACT FROM AUTHOR]

We study the Poincaré series of modules over a fiber product of commutative local rings. We introduce the notion of a weak complete intersection ideal; these are the ideals with the property that every differential in their minimal free resolutions can be represented by a matrix whose entries are in the ideal itself. We show that many properties of the Poincaré series that are known to hold for a fiber product over the maximal ideal also hold for those over weak intersection ideals. [ABSTRACT FROM AUTHOR]

In this paper, we consider whether existence of a sums-of-squares formula depends on the base field. We reformulate the question of existence as a question in algebraic geometry. We show that, for large enough p , existence of sums-of-squares formulas over algebraically closed fields is independent of the characteristic. We make the bound on p explicit, and we prove that the existence of a sums-of-squares formula of fixed type over an algebraically closed field is theoretically (though not practically) computable. [ABSTRACT FROM AUTHOR]

Let G be a finite group and K be a field of characteristic zero. Our purpose is to investigate the ideals of the slice Burnside functor K Ξ . It turns out that they are the subfunctors F of K Ξ such that for any finite group G , the evaluation F ( G ) is an ideal of the algebra K Ξ ( G ) . This allows for a determination of the full lattice of ideals of the slice Burnside p -biset functor K Ξ p . [ABSTRACT FROM AUTHOR]

In [10] , Terao proved the freeness of multi-Coxeter arrangements with constant multiplicities by giving an explicit construction of bases. Combining it with algebro-geometric method, Yoshinaga proved the freeness of the extended Catalan or Shi arrangements in [12] . However, there have been no explicit constructions of the bases for the logarithmic derivation modules of the extended Catalan and Shi arrangements. In this paper, we give the first explicit construction of them when the root system is of the type A 2 . [ABSTRACT FROM AUTHOR]

Let φ : S = k [ y 0 , . . . , y n ] → R = k [ y 0 , . . . , y n ] be given by y i → f i where f 0 , . . . , f n is an R -regular sequence of homogeneous elements of the same degree. A recent paper shows for ideals, I Δ ⊆ S , of matroids, Δ, that I Δ ( m ) ⊆ I r if and only if φ ⁎ ( I Δ ) ( m ) ⊆ φ ⁎ ( I Δ ) r where φ ⁎ ( I Δ ) is the ideal generated in R by φ ( I Δ ) . We prove this result for saturated homogeneous ideals I of configurations of points in P n and use it to obtain many new counterexamples to I ( r n − n + 1 ) ⊆ I r from previously known counterexamples. [ABSTRACT FROM AUTHOR]