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We study the relationship between antipodes on a Hopf algebroid ▪ in the sense of Böhm–Szlachanyi and the group of twists that lies inside the associated convolution algebra. We specialize to the case of a faithfully flat H -Hopf–Galois extensions B ⊆ A and related Ehresmann–Schauenburg bialgebroid. In particular, we find that the twists are in one-to-one correspondence with H -comodule algebra automorphism of A. We work out in detail the U (1) -extension O (C P q n − 1) ⊆ O (S q 2 n − 1) on the quantum projective space and show how to get an antipode on the bialgebroid out of the K -theory of the base algebra O (C P q n − 1). [ABSTRACT FROM AUTHOR]

We study the asymptotic non-vanishing of syzygies for products of projective spaces. Generalizing the monomial methods of Ein, Erman, and Lazarsfeld we give an explicit range in which the graded Betti numbers of P n 1 × P n 2 embedded by O P n 1 × P n 2 (d 1 , d 2) are non-zero. [ABSTRACT FROM AUTHOR]

In our previous work with Grifo and Hà, we showed the stable Harbourne-Huneke containment and Chudnovsky's conjecture for the defining ideal of sufficiently many general points in P N. In this paper, we establish the conjectures for all remaining cases, and hence, give the affirmative answer to Harbourne-Huneke containment and Chudnovsky's conjecture for the general points in P N for all N. Our new technique is to develop the Cremona reduction process that provides effective lower bounds for the Waldschmidt constant of the defining ideals of generic points in projective spaces. [ABSTRACT FROM AUTHOR]

Given the n -dimensional affine space over an arbitrary commutative ring k , G a group scheme flat and finitely presented over k and X ⊂ A k n a G -invariant affine and closed subscheme, we prove that the GIT quotient ▪ is of finite type over k , even if k is not noetherian, provided G is linearly reductive. This is well-known if ▪ is faithfully flat, which does not hold in general. We also explore the infinite-dimensional case. Concretely, we consider a G − k finitely presented projective module M and an arbitrary k -module N. We prove, under certain conditions on k and G , that the degrees of the generators of (S • (M ∨ ⊗ N)) G and the degrees of the generators of the ideal of relations are bounded. We encode this property into the notion of partially generated graded (pgg) algebra and we give their main properties. In particular, we prove the existence of canonical equivariant immersions of spectra of pgg algebras in certain projective spaces. [ABSTRACT FROM AUTHOR]

We construct polarized Calabi–Yau 3-folds with at worst isolated canonical orbifold points in codimension 4 that can be described in terms of the equations of the Segre embedding of P 2 × P 2 in P 8. We investigate the existence of other deformation families in their Hilbert scheme by either studying Tom and Jerry degenerations or by comparing their Hilbert series with those of existing low codimension Calabi–Yau 3-folds. Among other interesting results, we find a family of Calabi–Yau 3-fold with five distinct Tom and Jerry deformation families, a phenomenon not seen for Q -Fano 3-folds. We compute the Hodge numbers of P 2 × P 2 Calabi–Yau 3-folds and corresponding manifolds obtained by performing crepant resolutions. We obtain a manifold with a pair of Hodge numbers that does not appear in the famously known list of 30108 distinct Hodge pairs of Kruzer–Skarke, in the list of 7890 distinct Hodge pairs corresponding to complete intersections in the product of projective spaces and in Hodge paris obtained from Calabi–Yau 3-folds having low codimension embeddings in weighted projective spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the geometry of points in complex projective space that satisfy the Cayley–Bacharach condition with respect to the complete linear system of hypersurfaces of given degree. In particular, we improve a result by Lopez and Pirola and we show that, if k ≥ 1 and Γ = { P 1 , ... , P d } ⊂ P n is a set of distinct points satisfying the Cayley–Bacharach condition with respect to | O P n (k) | , with d ≤ h (k − h + 3) − 1 and 3 ≤ h ≤ 5 , then Γ lies on a curve of degree h − 1. Then we apply this result to the study of linear series on curves on smooth surfaces in P 3. Moreover, we discuss correspondences with null trace on smooth hypersurfaces of P n and on codimension 2 complete intersections. [ABSTRACT FROM AUTHOR]

We examine virtual resolutions of Stanley–Reisner ideals for a product of projective spaces by providing sufficient conditions for a simplicial complex to be virtually Cohen–Macaulay (to have a virtual resolution with length equal to its codimension). In particular, we show that all balanced simplicial complexes are virtually Cohen–Macaulay. [ABSTRACT FROM AUTHOR]

Let K be a complete discrete valued field with residue field k and F the function field of a curve over K. Let A ∈ 2 B r (F) be a central simple algebra with an involution σ of any kind and F 0 = F σ. Let h be an hermitian space over (A , σ) and G = S U (A , σ , h) if σ is of first kind and G = U (A , σ , h) if σ is of second kind. Suppose that char (k) ≠ 2 and ind (A) ≤ 4. Then we prove that projective homogeneous spaces under G over F 0 satisfy a local-global principle for rational points with respect to discrete valuations of F. [ABSTRACT FROM AUTHOR]

Euler-symmetric projective varieties, introduced by Baohua Fu and Jun-Muk Hwang in 2020, are nondegenerate projective varieties admitting many C × -actions of Euler type. They are quasi-homogeneous and uniquely determined by their fundamental forms at a general point. In this paper, we study complete intersections in projective spaces which are Euler-symmetric. It is proven that such varieties are complete intersections of hyperquadrics and the base locus of the second fundamental form at a general point is again a complete intersection. [ABSTRACT FROM AUTHOR]

We construct algebraic-geometry codes by using projective systems from projective curves over a finite field and the global sections of invertible sheaves on these curves. We also prove a formula for the Hilbert function of a finite set of points in a projective space in terms of the rank of a matrix constructed with the Veronese embedding and we use it to estimate the minimum distance of the dual codes. [ABSTRACT FROM AUTHOR]

Bonini, Borello and Byrne started the study of saturating linear sets in Desarguesian projective spaces, in connection with the covering problem in the rank metric. In this paper we study 1 -saturating linear sets in PG (2 , q 4) , that is F q -linear sets in PG (2 , q 4) with the property that their secant lines cover the entire plane. By making use of a characterization of generalized Gabidulin codes, we prove that the rank of such a linear set is at least 5. This answers to a recent question posed by Bartoli, Borello and Marino. [ABSTRACT FROM AUTHOR]

We introduce generalized Hessenberg varieties and establish basic facts. We show that the Tymoczko action of the symmetric group S n on the cohomology of Hessenberg varieties extends to generalized Hessenberg varieties and that natural morphisms among them preserve the action. By analyzing natural morphisms and birational maps among generalized Hessenberg varieties, we give an elementary proof of the Shareshian-Wachs conjecture. Moreover we present a natural generalization of the Shareshian-Wachs conjecture that involves generalized Hessenberg varieties and provide an elementary proof. As a byproduct, we propose a generalized Stanley-Stembridge conjecture for weighted graphs. Our investigation into the birational geometry of generalized Hessenberg varieties enables us to modify them into much simpler varieties like projective spaces or permutohedral varieties by explicit sequences of blowups or projective bundle maps. Using this, we provide two algorithms to compute the S n -representations on the cohomology of generalized Hessenberg varieties. As an application, we compute representations on the low degree cohomology of some Hessenberg varieties. [ABSTRACT FROM AUTHOR]

We show that the noncommutative differential geometry of quantum projective spaces is compatible with Rieffel's theory of compact quantum metric spaces. This amounts to a detailed investigation of the Connes metric coming from the unital spectral triple introduced by D'Andrea and Dąbrowski. In particular, we establish that the Connes metric metrizes the weak-⁎ topology on the state space of quantum projective space. This generalizes previous work by the second author and Aguilar regarding spectral metrics on the standard Podleś spheres. [ABSTRACT FROM AUTHOR]

Symmetric function theory is a key ingredient in the Schubert calculus of Grassmannians. Quasisymmetric functions are analogues that are similarly central to algebraic combinatorics, but for which the associated geometry is poorly developed. Baker and Richter (2008) showed that QSym manifests topologically as the cohomology ring of the loop suspension of infinite projective space or equivalently of its combinatorial homotopy model, the James reduced product J C P ∞. In recent work, we used this viewpoint to develop topologically-motivated bases of QSym and initiate a Schubert calculus for J C P ∞ in both cohomology and K -theory. Here, we study the torus-equivariant cohomology of J C P ∞. We identify a cellular basis and introduce double monomial quasisymmetric functions as combinatorial representatives, analogous to the factorial Schur functions and double Schubert polynomials of classical Schubert calculus. We also provide a combinatorial Littlewood–Richardson-type rule for the structure coefficients of this basis. Furthermore, we introduce an algebro-geometric analogue of the James reduced product construction. In particular, we prove that the James reduced product of a complex (quasi-)projective variety also carries the structure of a (quasi-)projective variety. [ABSTRACT FROM AUTHOR]

We address a natural question in noncommutative geometry, namely the rigidity observed in many examples, whereby noncommutative spaces (or equivalently their coordinate algebras) have very few automorphisms by comparison with their commutative counterparts. In the framework of noncommutative projective geometry, we define a groupoid whose objects are noncommutative projective spaces of a given dimension and whose morphisms correspond to isomorphisms of these. This groupoid is then a natural generalization of an automorphism group. Using work of Zhang, we may translate this structure to the algebraic side, wherein we consider homogeneous coordinate algebras of noncommutative projective spaces. The morphisms in our groupoid precisely correspond to the existence of a Zhang twist relating the two coordinate algebras. We analyse this automorphism groupoid, using the geometry of the point scheme, as introduced by Artin-Tate-Van den Bergh, to relate morphisms in our groupoid to certain automorphisms of the point scheme. We apply our results to two important examples, the two-dimensional quantum projective space and Sklyanin algebras. In both cases, we are able to use the geometry of the point schemes to fully describe the corresponding component of the automorphism groupoid. This provides a concrete description of the collection of Zhang twists of these algebras. [ABSTRACT FROM AUTHOR]

A clutter is clean if it has no delta or the blocker of an extended odd hole minor, and it is tangled if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the τ = 2 Conjecture. Let C be a clean tangled clutter. It was recently proved that C has a fractional packing of value two. Collecting the supports of all such fractional packings, we obtain what is called the core of C. The core is a duplication of the cuboid of a set of 0 − 1 points, called the setcore of C. In this paper, we prove three results about the setcore. First, the convex hull of the setcore is a full-dimensional polytope containing the center point of the hypercube in its interior. Secondly, this polytope is a simplex if, and only if, the setcore is the cocycle space of a projective geometry over the two-element field. Finally, if this polytope is a simplex of dimension more than three, then C has the clutter of the lines of the Fano plane as a minor. Our results expose a fascinating interplay between the combinatorics and the geometry of clean tangled clutters. [ABSTRACT FROM AUTHOR]

Given integers a 1 , a 2 , a 3 , there is a complex rank 3 topological bundle on C P 5 with i -th Chern class equal to a i if and only if a 1 , a 2 , a 3 satisfy the Schwarzenberger condition. Provided that the Schwarzenberger condition is satisfied, we prove that the number of isomorphism classes of rank 3 bundles V on C P 5 with c i (V) = a i is equal to 3 if a 1 and a 2 are both divisible by 3 and equal to 1 otherwise. This shows that Chern classes are incomplete invariants of topological rank 3 bundles on C P 5. To address this problem, we produce a universal class in the tm f (3) -cohomology of a Thom spectrum related to B U (3) , where tm f (3) denotes topological modular forms localized at 3. From this class and orientation data, we construct a Z / 3 -valued invariant of the bundles of interest and prove that our invariant separates distinct bundles with the same Chern classes. [ABSTRACT FROM AUTHOR]

Stability results play an important role in Galois geometries. The famous resultant method, developed by Szőnyi and Weiner [12] , [11] , became very fruitful and resulted in many stability theorems in the last two decades. This method is based on some bivariate polynomials associated to point sets. In this paper we generalize the method for the multidimensional case and show some new applications. We build up the multivariate polynomial machinery and apply it for Rédei polynomials. We can prove a high dimensional analogue of the result of Szőnyi-Weiner [9] , concerning the number of hyperplanes being skew to a point set of the space. We prove general results on "partial blocking sets", using the tools we have developed. [ABSTRACT FROM AUTHOR]

It is known that a Bruen chain of the three-dimensional projective space PG (3 , q) exists for every odd prime power q at most 37, except for q = 29. It was shown by Cardinali et al. (2005) that Bruen chains do not exist for 41 ⩽ q ⩽ 49. We develop a model, based on finite fields, which allows us to extend this result to 41 ⩽ q ⩽ 97 , thereby adding more evidence to the conjecture that Bruen chains do not exist for q > 37. Furthermore, we show that Bruen chains can be realised precisely as the (q + 1) / 2 -cliques of a two related, yet distinct, undirected simple graphs. [ABSTRACT FROM AUTHOR]

We prove that the space ℓ 2 – and more generally every infinite dimensional Banach space with an unconditional Schauder basis – can be renormed to have a Daugavet point. As a starting point, we introduce a new diametral notion for points of the unit sphere of Banach spaces, that complements the notion of Δ-points, but is weaker than the notion of Daugavet points. We show that this notion can be used to provide another geometric characterization of the Daugavet property, as well as to recover – and even to provide new – results about Daugavet points in various contexts such as absolute sums of Banach spaces or projective tensor products. Then, we observe that the study of this notion naturally leads to new renorming ideas in the context, and combine these with previous techniques from the literature to show that Daugavet points may exist even in super-reflexive spaces. [ABSTRACT FROM AUTHOR]

We give a new proof of a result of DiPippo and Wan for counting points of bounded height on projective spaces over global function fields. The new proof adapts the geometry of numbers arguments used by Schanuel in the number field case. [ABSTRACT FROM AUTHOR]

Let k be a perfect field and let GW (k) be the Grothendieck-Witt ring of (virtual) non-degenerate symmetric bilinear forms over k. We develop methods for computing the quadratic Euler characteristic χ (X / k) ∈ GW (k) for X a smooth hypersurface in a projective space and in a weighted projective space. We raise the question of a quadratic refinement of classical conductor formulas and find such a formula for the degeneration of a smooth hypersurface X in P n + 1 to the cone over a smooth hyperplane section of X ; we also find a similar formula in the weighted homogeneous case. We formulate a conjecture for similar types of degenerations, and we interpret the quadratic conductor formulas in terms of Ayoub's motivic nearby cycles functor. [ABSTRACT FROM AUTHOR]

Starting with the ground-breaking work of Cook II, Harbourne, Migliore and Nagel, there has been a lot of interest in unexpected hypersurfaces. In the last couple of months a considerable number of new examples and new phenomena has been observed and reported on. All examples studied so far had just one fat point. In this note we introduce a new series of examples, which establishes for the first time the existence of unexpected hypersurfaces with multiple fat points. The key underlying idea is to study Fermat-type configurations of points in projective spaces. [ABSTRACT FROM AUTHOR]

A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is 3-edge-colourable, the rest of cubic graphs fall into two classes: those that can be covered with four perfect matchings, and those that need at least five. Cubic graphs that require more than four perfect matchings to cover their edges are particularly interesting as potential counterexamples to several profound and long-standing conjectures including the celebrated cycle double cover conjecture. However, so far they have been extremely difficult to find. In this paper we build a theory that describes coverings with four perfect matchings as flows whose flow values represent points and outflow patterns represent lines of a configuration of ten points and six lines spanned by four points of the 3-dimensional projective space P 3 (F 2) in general position. This theory provides powerful tools for investigation of graphs that do not admit such a cover and offers a great variety of methods for their construction. As an illustrative example we produce a rich family of snarks (nontrivial cubic graphs with no 3-edge-colouring) that cannot be covered with four perfect matchings. The family contains all previously known graphs with this property. [ABSTRACT FROM AUTHOR]

We provide algorithms to reconstruct rational ruled surfaces in three-dimensional projective space from the "apparent contour" of a single projection to the projective plane. We deal with the case of tangent developables and of general projections to P 3 of rational normal scrolls. In the first case, we use the fact that every such surface is the projection of the tangent developable of a rational normal curve, while in the second we start by reconstructing the rational normal scroll. In both instances we then reconstruct the correct projection to P 3 of these surfaces by exploiting the information contained in the singularities of the apparent contour. [ABSTRACT FROM AUTHOR]

Given an embedding of a smooth projective curve X of genus g ≥ 1 into P N , we study the locus of linear subspaces of P N of codimension 2 such that projection from said subspace, composed with the embedding, gives a Galois morphism X → P 1. For genus g ≥ 2 we prove that this locus is a smooth projective variety with components isomorphic to projective spaces. If g = 1 and the embedding is given by a complete linear system, we prove that this locus is also a smooth projective variety whose positive-dimensional components are isomorphic to projective bundles over étale quotients of the elliptic curve, and we describe these components explicitly. [ABSTRACT FROM AUTHOR]

Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this paper, we consider the recently introduced linear version of such sets, which is, in turn, related to the covering problem in the rank metric. The main questions in this context are how small the rank of a saturating linear set can be and how to construct saturating linear sets of small rank. Recently, Bonini, Borello, and Byrne provided a lower bound on the rank of saturating linear sets in a given projective space, which is shown to be tight in some cases. In this paper, we provide construction of saturating linear sets meeting the lower bound and we develop a link between the saturating property and the scatteredness of linear sets. The last part of the paper is devoted to show some parameters for which the bound is not tight. [ABSTRACT FROM AUTHOR]

We consider polynomials expressing the cohomology classes of subvarieties of products of projective spaces, and limits of positive real multiples of such polynomials. We study the relation between these covolume polynomials and Lorentzian polynomials. While these are distinct notions, we prove that, like Lorentzian polynomials, covolume polynomials have M-convex support and generalize the notion of log-concave sequences. In fact, we prove that covolume polynomials are 'sectional log-concave', that is, the coefficients of suitable restrictions of these polynomials form log-concave sequences. We observe that Chern classes of globally generated bundles give rise to covolume polynomials, and use this fact to prove that certain polynomials associated with Segre classes of subschemes of products of projective spaces are covolume polynomials. We conjecture that the same polynomials may be Lorentzian after a standard normalization operation. Finally, we obtain a combinatorial application of a particular case of our Segre class result. We prove that the adjoint polynomial of a convex polyhedral cone contained in the nonnegative orthant, and sharing a face with it, is a covolume polynomial. This implies that these adjoint polynomials are M-convex and sectional log-concave, and in fact dually Lorentzian, that is, Lorentzian after a certain change of variables. [ABSTRACT FROM AUTHOR]

Explicit bases for the subfield subcodes of projective Reed-Muller codes over the projective plane and their duals are obtained. In particular, we provide a formula for the dimension of these codes. For the general case over the projective space, we generalize the necessary tools to deal with this case as well: we obtain a universal Gröbner basis for the vanishing ideal of the set of standard representatives of the projective space and we show how to reduce any monomial with respect to this Gröbner basis. With respect to the parameters of these codes, by considering subfield subcodes of projective Reed-Muller codes we obtain long linear codes with good parameters over a small finite field. [ABSTRACT FROM AUTHOR]

Pseudo-geometric designs are combinatorial designs which share the same parameters as a finite geometry design, but are not isomorphic to that design. As far as we know, many pseudo-geometric designs have been constructed by the methods of finite geometries and combinatorics. However, none of pseudo-geometric designs with the parameters S (2 , q + 1 , (q n − 1) / (q − 1)) is constructed by the approach of coding theory. In this paper, we use cyclic codes to construct pseudo-geometric designs. We firstly present a family of ternary cyclic codes from the m -sequences with Welch-type decimation d = 2 ⋅ 3 (n − 1) / 2 + 1 , and obtain some infinite family of 2-designs and a family of Steiner systems S (2 , 4 , (3 n − 1) / 2) using these cyclic codes and their duals. Moreover, the parameters of these cyclic codes and their shortened codes are also determined. Some of those ternary codes are optimal or almost optimal. Finally, we show that one of these obtained Steiner systems is inequivalent to the point-line design of the projective space PG (n − 1 , 3) and thus is a pseudo-geometric design. [ABSTRACT FROM AUTHOR]

We consider the problem of constructing matrices of linear forms of constant rank by focusing on the associated vector bundles on projective spaces. Important examples are given by the classical Steiner bundles, as well as some special (duals of) syzygy bundles that we call Drézet bundles. Using the classification of globally generated vector bundles with small first Chern class on projective spaces, we are able to describe completely the indecomposable matrices of constant rank up to six; some of them come from rigid homogeneous vector bundles, some other from Drézet bundles related either to plane quartics or to instanton bundles on P 3. [ABSTRACT FROM AUTHOR]

We introduce the weighted greatest common divisor of a tuple of integers and explore some of its basic properties. Furthermore, for a set of weights w = (q 0 , ... , q n) , we use the concept of the weighted greatest common divisor to define a height h (p) on weighted projective spaces WP w n (k). We prove some of the basic properties of this weighted height, including an analogue of the Northcott's theorem for heights on projective spaces. For a video summary of this paper, please visit https://youtu.be/Uh3BcwM2KJs. [ABSTRACT FROM AUTHOR]

We present a powerful method for the calculation of heat-kernel coefficients of the Laplacian on the projective complex spaces endowed with the Fubini-Study metric, and also for the Laplacian on the n -spheres equipped with the standard metric. Formulas for the regularized determinants are also given. Our formula in the case of the n -sphere recovers the result of Choi and Quine. [ABSTRACT FROM AUTHOR]

We give a complete classification of isomorphism classes of finitely generated projective modules, or equivalently, unitary equivalence classes of projections, over the C*-algebra C (S q 2 n + 1) of the quantum homogeneous sphere S q 2 n + 1. Then we explicitly identify as concrete elementary projections the quantum line bundles L k over the quantum complex projective space C P q n associated with the quantum Hopf principal U (1) -bundle S q 2 n + 1 → C P q n. [ABSTRACT FROM AUTHOR]

A biased graph is a graph with a class of selected circles ("cycles", "circuits"), called balanced, such that no theta subgraph contains exactly two balanced circles. A biased graph Ω has two natural matroids, the frame matroid G (Ω) and the lift matroid L (Ω) , and their extensions the full frame matroid G • (Ω) and the extended (or complete) lift matroid L 0 (Ω). In Part IV we used algebra to study the representations of these matroids by vectors over a skew field and the corresponding embeddings in Desarguesian projective spaces. Here we redevelop those representations, independently of Part IV and in greater generality, by using synthetic geometry. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study codimension one foliations on rational homogeneous spaces, with a focus on the moduli space of foliations of low degree on Grassmannians and cominuscule spaces. Using equivariant techniques, we show that codimension one degree zero foliations on (ordinary, orthogonal, symplectic) Grassmannians of lines, some spinor varieties, some Lagrangian Grassmannians, the Cayley plane (an E 6 -variety) and the Freudenthal variety (an E 7 -variety) are identified with restrictions of foliations on the ambient projective space. We also provide some evidence that such results can be extended beyond these cases. [ABSTRACT FROM AUTHOR]

Let L be a Galois algebra with Galois group G and let x be a normal element of L. The moduli space X of pairs (L , x) is isomorphic to an open subset of the quotient variety P / G , where P is the projective space of the regular representation of G. We provide a formula for the height of any pair (L , x) ∈ X (Q) in terms of algebraic invariants of L and x with respect to a natural adelic metric on the anticanonical divisor of P / G. [ABSTRACT FROM AUTHOR]

The starting point of this paper is a duality for sequences of natural numbers which, under mild hypotheses, interchanges subadditive and superadditive sequences and inverts their asymptotic growth constants. We are motivated to explore this sequence duality since it arises naturally in at least two important algebraic-geometric contexts. The first context is Macaulay-Matlis duality, where the sequence of initial degrees of the family of symbolic powers of a radical ideal is dual to the sequence of Castelnuovo-Mumford regularity values of a quotient by ideals generated by powers of linear forms. This philosophy is drawn from an influential paper of Emsalem and Iarrobino. We generalize this duality to differentially closed graded filtrations of ideals. In a different direction, we establish a duality between the sequence of Castelnuovo-Mumford regularity values of the symbolic powers of certain ideals and a geometrically inspired sequence we term the jet separation sequence. We show that this duality underpins the reciprocity between two important geometric invariants: the multipoint Seshadri constant and the asymptotic regularity of a set of points in projective space. [ABSTRACT FROM AUTHOR]

We give a quantum version of the Danilov-Jurkiewicz presentation of the cohomology of a compact toric orbifold with projective coarse moduli space. More precisely, we construct a canonical isomorphism from a formal version of the Batyrev ring from [4] to the quantum orbifold cohomology at a canonical bulk deformation. This isomorphism generalizes results of Givental [23] , Iritani [34] and Fukaya-Oh-Ohta-Ono [21] for toric manifolds and Coates-Lee-Corti-Tseng [11] for weighted projective spaces. The proof uses a quantum version of Kirwan surjectivity (Theorem 2.6 below) and an equality of dimensions (Theorem 4.19 below) deduced using a toric minimal model program (tmmp). We show that there is a natural decomposition of the quantum cohomology where summands correspond to singularities in the tmmp, each of which gives rise to a collection of Hamiltonian non-displaceable Lagrangian tori. [ABSTRACT FROM AUTHOR]

We determine the maximum number of F q -rational points that a nonsingular threefold of degree d in a projective space of dimension 4 defined over F q may contain. This settles a conjecture by Homma and Kim concerning the maximum number of points on a hypersurface in a projective space of even dimension in this particular case. [ABSTRACT FROM AUTHOR]

Abstract In this paper we prove an Erdős-Ko-Rado-type theorem for intersecting sets of permutations. We show that an intersecting set of maximal size in the projective general linear group PGL n + 1 (q) , in its natural action on the points of the n -dimensional projective space, is either a coset of the stabiliser of a point or a coset of the stabiliser of a hyperplane. This gives a positive solution to [15, Conjecture 2]. [ABSTRACT FROM AUTHOR]

We construct log resolutions of pairs on the blow-up of the projective space in an arbitrary number of general points and we discuss the semi-ampleness of the strict transforms. As an application we give an explicit proof that the abundance conjecture holds for an infinite family of such pairs. For n + 2 points in P n , these strict transforms are F-nef divisors on the moduli space M ‾ 0 , n + 3 in a Kapranov's model: we show that all of them are nef. [ABSTRACT FROM AUTHOR]

Abstract We introduce a theory of multigraded Cayley-Chow forms associated to subvarieties of products of projective spaces. Two new phenomena arise: first, the construction turns out to require certain inequalities on the dimensions of projections; and second, in positive characteristic the multigraded Cayley-Chow forms can have higher multiplicities. The theory also provides a natural framework for understanding multifocal tensors in computer vision. [ABSTRACT FROM AUTHOR]

Abstract We apply Orlov's equivalence to derive formulas for the Betti numbers of maximal Cohen–Macaulay modules over the cone an elliptic curve (E , x) embedded into P n − 1 , by the full linear system | O (n x) | , for n > 3. The answers are given in terms of recursive sequences. These results are applied to give a criterion of (Co-)Koszulity. In the last two sections of the paper we apply our methods to study the cases n = 1 , 2. Geometrically these correspond to the embedding of an elliptic curve into a weighted projective space. The singularities of the corresponding cones are called minimal elliptic. They were studied by K. Saito [13] , where he introduced the notation E 8 ˜ for n = 1 , E 7 ˜ for n = 2 and E 6 ˜ for the cone over a smooth cubic, that is, for the case n = 3. For the singularities E 7 ˜ and E 8 ˜ we obtain formulas for the Betti numbers and the numerical invariants of MCM modules analogous to the case of a plane cubic. [ABSTRACT FROM AUTHOR]

Abstract We explicitly bound T-singularities on normal projective surfaces W with one singularity, and K W ample. This bound depends only on K W 2 , and it is optimal when W is not rational. We classify and realize surfaces attaining the bound for each nonnegative Kodaira dimension of the minimal resolution of W. This answers effectiveness of bounds (see [1] , [2] , [17]) for those surfaces. [ABSTRACT FROM AUTHOR]

Abstract We study compactifications of Drinfeld half-spaces over a finite field. In particular, we construct a purely inseparable endomorphism of Drinfeld's half-space Ω (V) over a finite field k that does not extend to an endomorphism of the projective space P (V). This should be compared with theorem of Rémy, Thuillier and Werner that every k -automorphism of Ω (V) extends to a k -automorphism of P (V). Our construction uses an inseparable analogue of the Cremona transformation. We also study foliations on Drinfeld's half-spaces. This leads to various examples of interesting varieties in positive characteristic. In particular, we show a new example of a non-liftable projective Calabi–Yau threefold in characteristic 2 and we show examples of rational surfaces with klt singularities, whose cotangent bundle contains an ample line bundle. [ABSTRACT FROM AUTHOR]

Abstract We provide a common description and geometric characterization of the Veronesean representations of all projective spaces defined over finite-dimensional quadratic alternative division algebras. We also study the homogeneity of these representations, and the irreducibility and indecomposability of the induced projective representations of the corresponding simple (automorphism) groups. [ABSTRACT FROM AUTHOR]

Abstract In [2] and [18] are presented the first two families of maximum scattered F q -linear sets of the projective line PG (1 , q n). More recently in [22] and in [5] , new examples of maximum scattered F q -subspaces of V (2 , q n) have been constructed, but the equivalence problem of the corresponding linear sets is left open. Here we show that the F q -linear sets presented in [22] and in [5] , for n = 6 , 8 , are new. Also, for q odd, q ≡ ± 1 , 0 (mod 5) , we present new examples of maximum scattered F q -linear sets in PG (1 , q 6) , arising from trinomial polynomials, which define new F q -linear MRD-codes of F q 6 × 6 with dimension 12, minimum distance 5 and left idealiser isomorphic to F q 6 . [ABSTRACT FROM AUTHOR]

In this paper, we develop a variation of the index calculus algorithm using non-planar models of non-hyperelliptic curves of genus g. Using canonical model of degree 2 g − 2 in the projective space of dimension g − 1 , intersections with hyperplanes and following similar ideas to those of Diem (who used intersections with lines on planar models), we obtain an upper bound of O (q 2 − 2 g − 1 + ε) for the computation of discrete logarithms for all non-hyperelliptic curves of genus g defined over the finite field F q. This asymptotic cost is essentially the same as Diem's, but our algorithm offers several advantages over Diem's, including a constant speed-up. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]