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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]

Let f:(Cn,S)→(Cp,0)$f: (\mathbb {C}^n,S)\rightarrow (\mathbb {C}^p,0)$ be a finite map‐germ with n

K3 surfaces play a prominent role in string theory and algebraic geometry. The properties of their enumerative invariants have important consequences in black hole physics and in number theory. To a K3 surface, string theory associates an Elliptic genus, a certain partition function directly related to the theory of Jacobi modular forms. A multiplicative lift of the Elliptic genus produces another modular object, an Igusa cusp form, which is the generating function of BPS invariants of K3 × E . In this note, we will discuss a refinement of this chain of ideas. The Elliptic genus can be generalized to the so-called Hodge-Elliptic genus which is then related to the counting of refined BPS states of K3 × E . We show how such BPS invariants can be computed explicitly in terms of different versions of the Hodge-Elliptic genus, sometimes in closed form, and discuss some generalizations. [ABSTRACT FROM AUTHOR]

We report new examples of Sidon sets in abelian groups arising from generalized Jacobians of curves, and discuss some of their properties with respect to size and structure. [ABSTRACT FROM AUTHOR]

In this paper, we study the action of diamond operators on Hilbert modular forms with coefficients in a general commutative ring. In particular, we generalize a result of Chai on the surjectivity of the constant term map for Hilbert modular forms with nebentype to the setting of group ring valued modular forms. As an application, we construct certain Hilbert modular forms required for Dasgupta–Kakde's proof of the Brumer–Stark conjecture at odd primes. Since the forms required for the Brumer–Stark conjecture live on the non-PEL Shimura variety associated to the reductive group G = Res F / Q (GL 2) , as opposed to the PEL Shimura variety associated to the subgroup G ∗ ⊂ G studied by Chai, we give a detailed explanation of theory of algebraic diamond operators for G, as well as how the theory of toroidal and minimal compactifications for G may be deduced from the analogous theory for G ∗ . [ABSTRACT FROM AUTHOR]

We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K ⊆ ⋀ 2 V , where V is a vector space. Previously Koszul modules of finite length have been used to give a proof of Green's Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the instability of sufficiently positive rank 2 vector bundles on curves is governed by resonance and give a splitting criterion. [ABSTRACT FROM AUTHOR]

We give the explicit equations for a P 3 × P 3 embedding of the Jacobian of a curve of genus 2, which gives a natural analog for abelian surfaces of the Edwards curve model of elliptic curves. This gives a much more succinct description of the Jacobian variety than the standard version in P 15 . We also give a condition under which, as for the Edwards curve, the abelian surfaces have a universal group law. [ABSTRACT FROM AUTHOR]

Sheaves are mathematical objects that describe the globally compatible data associated with open sets of a topological space. Original examples of sheaves were continuous functions; later they also became powerful tools in algebraic geometry, as well as logic and set theory. More recently, sheaves have been applied to the theory of contextuality in quantum mechanics. Whenever the local data are not necessarily compatible, sheaves are replaced by the simpler setting of presheaves. In previous work, we used presheaves to model lexically ambiguous phrases in natural language and identified the order of their disambiguation. In the work presented here, we model syntactic ambiguities and study a phenomenon in human parsing called garden-pathing. It has been shown that the information-theoretic quantity known as 'surprisal' correlates with human reading times in natural language but fails to do so in garden-path sentences. We compute the degree of signalling in our presheaves using probabilities from the large language model BERT and evaluate predictions on two psycholinguistic datasets. Our degree of signalling outperforms surprisal in two ways: (i) it distinguishes between hard and easy garden-path sentences (with a p -value <10−5), whereas existing work could not, (ii) its garden-path effect is larger in one of the datasets (32 ms versus 8.75 ms per word), leading to better prediction accuracies. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'. [ABSTRACT FROM AUTHOR]

We define the notion of Y-algebroids, generalising the Lie, Courant, and exceptional algebroids that have been used to capture the local symmetry structure of type II string theory and M-theory compactifications to D ≥ 5 dimensions. Instead of an invariant inner product, or its generalisation arising in exceptional algebroids, Y-algebroids are built around a specific type of tensor, denoted Y , that provides exactly the necessary properties to also describe compactifications to D = 4 dimensions. We classify "M-exact" E7-algebroids and show that this precisely matches the form of the generalised tangent space of E7(7) × ℝ+-generalised geometry, with possible twists due to 1-, 4- and 7-form fluxes, corresponding physically to the derivative of the warp factor and the M-theory fluxes. We translate the notion of generalised Leibniz parallelisable spaces, relevant to consistent truncations, into this language, where they are mapped to so-called exceptional Manin pairs. We also show how to understand Poisson-Lie U-duality and exceptional complex structures using Y-algebroids. [ABSTRACT FROM AUTHOR]

A topological space is called self-covering if it is a nontrivial cover of itself. We prove that a closed self-covering manifold M with free abelian fundamental group fibers over a circle under mild assumptions. In particular, we give a complete answer to the question whether a self-covering manifold with fundamental group \mathbb Z is a fiber bundle over S^1, except for the 4-dimensional smooth case. As an algebraic Hilfssatz, we develop a criterion for finite generation of modules over a commutative Noetherian ring. We also construct examples of self-covering manifolds with nonfree abelian fundamental group, which are not fiber bundles over S^1. [ABSTRACT FROM AUTHOR]

We establish a connection between the algebraic geometry of the type B$B$ permutohedral toric variety and the combinatorics of delta‐matroids. Using this connection, we compute the volume and lattice point counts of type B$B$ generalized permutohedra. Applying tropical Hodge theory to a new framework of "tautological classes of delta‐matroids," modeled after certain vector bundles associated to realizable delta‐matroids, we establish the log‐concavity of a Tutte‐like invariant for a broad family of delta‐matroids that includes all realizable delta‐matroids. Our results include new log‐concavity statements for all (ordinary) matroids as special cases. [ABSTRACT FROM AUTHOR]

Using tools from algebraic topology and algebraic geometry (fundamental polygons and blowups, respectively), we prove that cutting along the center of a Möbius strip yields a cylinder. [ABSTRACT FROM AUTHOR]

Motivated by the study of the Kahan–Hirota–Kimura discretisation of the Euler top, we characterise the growth and integrability properties of a collection of elements in the Cremona group of a complex projective 3-space using techniques from algebraic geometry. This collection consists of maps obtained by composing the standard Cremona transformation c 3 ∈ Bir (P 3) with projectivities that permute the fixed points of c 3 and the points over which c 3 performs a divisorial contraction. Specifically, we show that three behaviour are possible: (A) integrable with quadratic degree growth and two invariants, (B) periodic with two-periodic degree sequences and more than two invariants, and (C) non-integrable with submaximal degree growth and one invariant. [ABSTRACT FROM AUTHOR]

We study the cone of completely positive (cp) matrices for the first interesting case n = 5 . This is a semialgebraic set for which the polynomial equalities and inequlities that define its boundary can be derived. We characterize the different loci of this boundary and we examine the two open sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and able to compute the cp-factorization even for matrices in the boundary. With our results, many new example cases can be produced and several insightful numerical experiments are performed that illustrate the difficulty of the cp-factorization problem. [ABSTRACT FROM AUTHOR]

We prove that if a symplectic 4-manifold X becomes a rational 4-manifold after applying rational blow-down surgery, then the symplectic 4-manifold X is originally rational. That is, a symplectic rational blow-up of a rational symplectic 4-manifold is again rational. As an application we show that a degeneration of a family of smooth rational complex surfaces is a rational surface if the degeneration has at most quotient surface singularities, which generalizes slightly a classical result of Bădescu [J. Reine Angew. Math. 367 (1986), pp. 76–89] in algebraic geometry under a mild additional condition. [ABSTRACT FROM AUTHOR]

By using the methods of Picard-Fuchs equation and Riccati equation, we study the upper bound of the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one under polynomial perturbations of degree n. We obtain that the upper bound is 7[(n - 3)/2] + 5 when n = 5, 8 when n = 4, 5 when n = 3, 4 when n = 2, and 0 when n = 1 or n = 0, which linearly depends on n. [ABSTRACT FROM AUTHOR]

We solve Prill's problem, originally posed by David Prill in the late 1970s and popularized in Arbarello, Cornalba, Griffiths and Harris's "Geometry of Algebraic Curves." That is, for any curve Y of genus 2, we produce a finite 'etale degree 36 connected cover f: X → Y where, for every point y ∈ Y, the preimage f-1(y) moves in a pencil. [ABSTRACT FROM AUTHOR]

In this paper we consider a family F of 2n-dimensional F q -linear rank metric codes in F q n × n arising from polynomials of the form x q s + δ x q n 2 + s ∈ F q n [ x ] . The family F was introduced by Csajbók et al. (JAMA 548:203–220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that F contains MRD codes for n = 8 , and other subsequent partial results have been provided in the literature towards the classification of MRD codes in F for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case n = 8 , providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional F q -rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far. [ABSTRACT FROM AUTHOR]

Algebra environments provide requisites for studying objects of interest in operator algebra theory, group representation theory, spin geometry, Clifford analysis, and several variable operator theory. The concept is analyzed by developing an algebraic geometry approach. Specific algebraic sets, called structure manifolds of algebra environments, and their Zariski tangent spaces are introduced and described by using as critical tools derivations on algebras. Structure manifolds of tensor environments in particular yield spaces of algebra homomorphisms. Consequently, such spaces could be investigated as algebraic manifolds. Related issues include characterizations of their Zariski tangent spaces and of derivations that preserve algebra homomorphisms. [ABSTRACT FROM AUTHOR]

Young diagrams are fundamental combinatorial objects in representation theory and algebraic geometry. Many constructions that rely on these objects depend on variations of a straightening process that expresses a filling of a Young diagram as a sum of semistandard tableaux subject to certain relations. This paper solves the long standing open problem of giving a non-iterative formula for straightening a filling. We apply our formula to give a complete generalization of a theorem of Gonciulea and Lakshmibai. [ABSTRACT FROM AUTHOR]

We equip integral graded-polarized mixed period spaces with a natural Ralg-definable analytic structure, and prove that any period map associated to an admissible variation of integral graded-polarized mixed Hodge structures is definable in Ran,exp with respect to this structure. As a consequence we re-prove that the zero loci of admissible normal functions are algebraic. [ABSTRACT FROM AUTHOR]

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]

We introduce a conjecture on homological mirror symmetry relating the symplectic topology of the complement of a smooth ample divisor in a K3 surface to the algebraic geometry of type III degenerations. We prove it when the degree of the divisor is either 2 or 4. [ABSTRACT FROM AUTHOR]

Poset-blowdown of subgroup posets of groups is an analog of blowdown in algebraic geometry. It is a poset map obtained by contracting normal subgroups. For finite groups, this is considered as a map between the Hasse diagrams of the subgroup posets. Poset-blowdowns are classified into three types: tame, wild, and hybrid depending on the sizes of their fibers. In this paper we describe the poset-blowdowns for generalized quaternion groups Q2n (n ≥ 3). They have distinguished nature in that all types (tame, wild, and hybrid) appear in the successive poset-blowdowns associated with the three chief series of Q2n. [ABSTRACT FROM AUTHOR]

In this paper, we consider the affine variety codes obtained evaluating the polynomials b y = a k x k + ... + a 1 x + a 0 , b , a i ∈ F q r , at the affine F q r -rational points of the Norm-Trace curve. In particular, we investigate the weight distribution and the set of minimal codewords. Our approach, which uses tools of algebraic geometry, is based on the study of the absolute irreducibility of certain algebraic varieties. [ABSTRACT FROM AUTHOR]

Motivated by their role in M‐theory, F‐theory, and S‐theory compactifications, all possible complete intersections Calabi‐Yau five‐folds in a product of four or less complex projective spaces are constructed, with up to four constraints. A total of 27 068 spaces are obtained, which are not related by permutations of rows and columns of the configuration matrix, and determine the Euler number for all of them. Excluding the 3909 product manifolds among those, the cohomological data for 12 433 cases are calculated, i.e., 53.7% of the non‐product spaces, obtaining 2375 different Hodge diamonds. The dataset containing all the above information is available here. The distributions of the invariants are presented, and a comparison with the lower‐dimensional analogues is discussed. Supervised machine learning is performed on the cohomological data, via classifier, and regressor (both fully connected and convolutional) neural networks. h1, 1 can be learnt very efficiently, with very high R2 score and an accuracy of 96% is found, i.e., 96% of the predictions exactly match the correct values. For h1,4,h2,3,η$h^{1,4},h^{2,3}, \eta$, very high R2 scores are also found, but the accuracy is lower, due to the large ranges of possible values. [ABSTRACT FROM AUTHOR]

We develop a new method that improves the efficiency of equation-by-equation homotopy continuation methods for solving polynomial systems. Our method is based on a novel geometric construction and reduces the total number of homotopy paths that must be numerically continued. These improvements may be applied to the basic algorithms of numerical algebraic geometry in the settings of both projective and multiprojective varieties. Our computational experiments demonstrate significant savings obtained on several benchmark systems. We also present an extended case study on maximum likelihood estimation for rank-constrained symmetric n × n matrices, in which multiprojective u-generation allows us to complete the list of ML degrees for n ≤ 6. [ABSTRACT FROM AUTHOR]

Kazhdan and Lusztig identified the affine Hecke algebra ℋ with an equivariant K -group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields F with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from K -theory to Hochschild homology and thereby identify ℋ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, the coherent Springer sheaf. As a result the derived category of ℋ-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu). In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of GL n (F) into coherent sheaves on the stack of Langlands parameters. [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]

The Hilbert scheme of points |$\textrm {Hilb}^{n}(S)$| of a smooth surface |$S$| is a well-studied parameter space, lying at the interface of algebraic geometry, commutative algebra, representation theory, combinatorics, and mathematical physics. The foundational result is a classical theorem of Fogarty, stating that |$\textrm {Hilb}^{n}(S)$| is a smooth variety of dimension |$2n$|⁠. In recent years there has been growing interest in a natural generalization of |$\textrm {Hilb}^{n}(S)$|⁠ , the nested Hilbert scheme |$\textrm {Hilb}^{(n_{1}, n_{2})}(S)$|⁠ , which parametrizes nested pairs of zero-dimensional subschemes |$Z_{1} \supseteq Z_{2}$| of |$S$| with |$\deg Z_{i}=n_{i}$|⁠. In contrast to Fogarty's theorem, |$\textrm {Hilb}^{(n_{1}, n_{2})}(S)$| is almost always singular, and very little is known about its singularities. In this paper, we aim to advance the knowledge of the geometry of these nested Hilbert schemes. Work by Fogarty in the 70's shows that |$\textrm {Hilb}^{(n,1)}(S)$| is a normal Cohen–Macaulay variety, and Song more recently proved that it has rational singularities. In our main result, we prove that the nested Hilbert scheme |$\textrm {Hilb}^{(n,2)}(S)$| has rational singularities. We employ an array of tools from commutative algebra to prove this theorem. Using Gröbner bases, we establish a connection between |$\textrm {Hilb}^{(n,2)}(S)$| and a certain variety of matrices with an action of the general linear group. This variety of matrices plays a central role in our work, and we analyze it by various algebraic techniques, including the Kempf–Lascoux–Weyman technique of calculating syzygies, square-free Gröbner degenerations, and the Stanley–Reisner correspondence. Along the way, we also obtain results on classes of irreducible and reducible nested Hilbert schemes, dimension of singular loci, and |$F$| -singularities in positive characteristic. [ABSTRACT FROM AUTHOR]

Let C be a linear code of length n and dimension k over the finite field F q m . The trace code Tr (C) is a linear code of the same length n over the subfield F q . The obvious upper bound for the dimension of the trace code over F q is m k. If equality holds, then we say that C has maximum trace dimension. The problem of finding the true dimension of trace codes and their duals is relevant for the size of the public key of various code-based cryptographic protocols. Let C a denote the code obtained from C and a multiplier vector a ∈ ( F q m ) n . In this study, we give a lower bound for the probability that a random multiplier vector produces a code C a of maximum trace dimension. We give an interpretation of the bound for the class of algebraic geometry codes in terms of the degree of the defining divisor. The bound explains the experimental fact that random alternant codes have minimal dimension. Our bound holds whenever n ≥ m (k + h) , where h ≥ 0 is the Singleton defect of C. For the extremal case n = m (h + k) , numerical experiments reveal a closed connection between the probability of having maximum trace dimension and the probability that a random matrix has full rank. [ABSTRACT FROM AUTHOR]

Govindan and Klumpp [7] provided a characterization of perfect equilibria using Lexicographic Probability Systems (LPSs). Their characterization was essentially finite in that they showed that there exists a finite bound on the number of levels in the LPS, but they did not compute it explicitly. In this note, we draw on two recent developments in Real Algebraic Geometry to obtain a formula for this bound. [ABSTRACT FROM AUTHOR]

The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions of affine topological spaces. The non-degenerate class maintains the degree preservation principle of the atoms of the polynomials of a topological simplex, which is relaxed in the degenerate class. The concept of hybrid decomposition of a simplicial polynomial in the non-degenerate class is introduced. The decompositions of simplicial polynomial for a large set of simplex vertices generate ideal components from the radical, and the components preserve the topologically isolated origin in all cases within the topological manifolds. Interestingly, the topological manifolds generated by a non-degenerate class of simplicial polynomials do not retain the homeomorphism property under polynomial extension by atom addition if the simplicial condition is violated. However, the topological manifolds generated by the degenerate class always preserve isomorphism with varying rotational orientations. The hybrid decompositions of the non-degenerate class of simplicial polynomials give rise to the formation of simplicial chains. The proposed formulations do not impose strict positivity on simplicial polynomials as a precondition. [ABSTRACT FROM AUTHOR]

What is striking about de Broglie's foundational work on wave–particle dualism is the role played by pseudo-Riemannian geometry in his early thinking. While exploring a fully covariant description of the Klein–Gordon equation, he was led to the revolutionary idea that a variable rest mass was essential. DeWitt later explained that in order to obtain a covariant quantum Hamiltonian, one must supplement the classical Hamiltonian with an additional energy ℏ 2 Q from which the quantum potential emerges, a potential that Berry has recently shown also arises in classical wave optics. In this paper, we show how these ideas emerge from an essentially geometric structure in which the information normally carried by the wave function is contained within the algebraic description of the geometry itself, within an element of a minimal left ideal. We establish the fundamental importance of conformal symmetry, in which rescaling of the rest mass plays a vital role. Thus, we have the basis for a radically new theory of quantum phenomena based on the process of mass-energy flow. [ABSTRACT FROM AUTHOR]

We obtain new bounds on short Weil sums over small multiplicative subgroups of prime finite fields which remain nontrivial in the range the classical Weil bound is already trivial. The method we use is a blend of techniques coming from algebraic geometry and additive combinatorics. [ABSTRACT FROM AUTHOR]

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini's theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties. [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]

This article, published in Mathematical Notes, discusses Hurwitz numbers for reflection groups. Hurwitz numbers are used to solve problems in combinatorics, topology, and algebraic geometry. The article explores the structure of the reflection groups and , and describes the conjugacy classes within these groups. It also discusses the relationship between Hurwitz numbers and cut-and-join equations. The article concludes with a discussion of the KP hierarchy and its connection to Hurwitz numbers. The research was funded by the National Research University Higher School of Economics and the International Laboratory of Cluster Geometry. [Extracted from the article]

There are two main problems in finding the higher genus superstring measure. The first one is that for g ≥ 5 the super moduli space is not projected. Furthermore, the supermeasure is regular for g ≤ 11, a bound related to the source of singularities due to the divisor in the moduli space of Riemann surfaces with even spin structure having holomorphic sections, such a divisor is called the θ-null divisor. A result of this paper is the characterization of such a divisor. This is done by first extending the Dirac propagator, that is the Szegö kernel, to the case of an arbitrary number of zero modes, that leads to a modification of the Fay trisecant identity, where the determinant of the Dirac propagators is replaced by the product of two determinants of the Dirac zero modes. By taking suitable limits of points on the Riemann surface, this holomorphic Fay trisecant identity leads to identities that include points dependent rank 3 quadrics in ℙg−1. Furthermore, integrating over the homological cycles gives relations for the Riemann period matrix which are satisfied in the presence of Dirac zero modes. Such identities characterize the θ-null divisor. Finally, we provide the geometrical interpretation of the above points dependent quadrics and show, via a new θ-identity, its relation with the Andreotti-Mayer quadric. [ABSTRACT FROM AUTHOR]

We present a method to construct the extended Kähler cone of any Calabi-Yau threefold by using Gopakumar-Vafa invariants to identify all geometric phases that are related by flops or Weyl reflections. In this way we obtain the Kähler moduli spaces of all favorable Calabi-Yau threefold hypersurfaces with h1,1 ≤ 4, including toric and non-toric phases. In this setting we perform an explicit test of the Weak Gravity Conjecture by using the Gopakumar-Vafa invariants to count BPS states. All of our examples satisfy the tower/sublattice WGC, and in fact they even satisfy the stronger lattice WGC. [ABSTRACT FROM AUTHOR]

By applying an inverse Kaluza-Klein procedure, we provide explicit coordinate expressions for Riemannian metrics on two homeomorphic but not diffeomorphic spheres in seven dimensions. We identify Milnor's bundles, among which ten out of the fourteen exotic seven-spheres appear (ignoring orientation), with non-principal bundles having homogeneous fibres. Then, we use the techniques in [1] to obtain a general ansatz for the coordinate expression of a metric on the total space of any Milnor's bundle. The ansatz is given in terms of a metric on S4, a metric on S3 (which can smoothly vary throughout S4), and a connection on the principal SO(4)-bundle over S4. As a concrete example, we present explicit formulae for such metrics for the ordinary sphere and the Gromoll-Meyer exotic sphere. Then, we perform a non-abelian Kaluza-Klein reduction to gravity in seven dimensions, according to (a slightly simplified version of) the metric ansatz above. We obtain the standard four-dimensional Einstein-Yang-Mills system, for which we find solutions associated with the geometries of the ordinary sphere and of the exotic one. The two differ by the winding numbers of the instantons involved. [ABSTRACT FROM AUTHOR]

State observers for nonlinear systems are often designed for a canonical form of this system. However, this form may possess singular points, where the vector field is not defined or a Lipschitz condition is not fulfilled. This unpleasant behavior can possibly be avoided using an embedding into a higher dimensional space. A construction of such an embedding and the corresponding inverse map is discussed for polynomial systems using methods from algebraic geometry. [ABSTRACT FROM AUTHOR]

The paper deals with a class of periods, Frobenius constants, which describe monodromy of Frobenius solutions of differential equations arising in algebraic geometry. We represent Frobenius constants related to families of elliptic curves as iterated integrals of modular forms. Using the theory of periods of modular forms, we then witness some of these constants in terms of zeta values. [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]

An MDS code is a code which achieves equality in the singleton bound. The defect of a code measures how far it is from an MDS code. Amplifying on the relationship between the weight distribution of a code and its dual code as in the well-known MacWilliams identities, we show in this paper that there are indeed strong lower bounds on the defects of codes or the dual codes. [ABSTRACT FROM AUTHOR]

We study the spectra of operators on periodic graphs using methods from combinatorial algebraic geometry. Our main result is a bound on the number of complex critical points of the Bloch variety, together with an effective criterion for when this bound is attained. We show that this criterion holds for Z²- and Z³-periodic graphs with sufficiently many edges and use our results to establish the spectral edges conjecture for some Z²-periodic graphs. [ABSTRACT FROM AUTHOR]