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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]

Given a smooth, projective curve Y, a finite group G and a positive integer n we study smooth, proper families X → Y x S → S of Galois covers of Y with Galois group isomorphic to G branched in n points, parameterized by algebraic varieties S. When G is with trivial center we prove that the Hurwitz space HnG (Y) is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary G we prove that HnG (Y) is a coarse moduli variety. For families of pointed Galois covers of (Y, y0) we prove that the Hurwitz space HnG (Y, y0) is a fine moduli variety, and construct explicitly the universal family, for arbitrary group G. We use classical tools of algebraic topology and of complex algebraic geometry. [ABSTRACT FROM AUTHOR]

We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. For elliptic normal curves, we further obtain definite symmetric determinantal representations for the hyperbolic secant hypersurfaces, which implies the existence of symmetric Ulrich sheaves of rank one on these hypersurfaces. We also use this to derive better bounds on the size of semidefinite representations for convex hulls of real algebraic curves of genus 1. [ABSTRACT FROM AUTHOR]

In algebraic geometry, the Frobenius map (or called Frobenius action , or Frobenius operator) F ⁎ on cohomology groups plays an important role in the classification of algebraic varieties over a field of positive characteristic. In particular, representation matrices for F ⁎ give rise to many important invariants such as p -rank and a -number. Several methods for computing representation matrices for F ⁎ have been proposed for specific curves. In this paper, we present an algorithm to compute representation matrices for F ⁎ of general projective schemes over a perfect field of positive characteristic. We also propose an efficient algorithm specific to complete intersections; it requires to compute only certain coefficients in a power of a multivariate polynomial. Our algorithms shall derive fruitful applications such as computing Hasse-Witt matrices, and enumerating superspecial curves. In particular, the second algorithm for complete intersections provides a useful tool to judge the superspeciality of an algebraic curve, which is a key ingredient to prove main results in Kudo and Harashita (2017a,b, 2020) on the enumeration of superspecial genus-4 curves. [ABSTRACT FROM AUTHOR]

For n ∈ N, we consider the algebraic variety V obtained by intersecting n + 1algebraic curves of degree n in R², when the leading terms of the associated bivariate polynomials are all different. We provide a new proof, based on the Flat Extension Theorem from the theory of truncated moment problems, that the cardinality of V cannot exceed (2n+1). In some instances, this provides a slightly better estimate than the one given by Bézout's Theorem. Our main result contributes to the growing literature on the interplay between linear algebra, operator theory, and real algebraic geometry. [ABSTRACT FROM AUTHOR]

Rapoport–Zink spaces, or more generally local Shimura varieties, are expected to provide geometric realization of the local Langlands correspondence via their l‐adic cohomology. Along this line is a conjecture by Harris and Viehmann, which roughly says that when the underlying local Shimura datum is not basic, the l‐adic cohomology of the local Shimura variety is parabolically induced. We verify this conjecture for Rapoport–Zink spaces which are Hodge type and Hodge–Newton reducible. The main strategy is to embed such a Rapoport–Zink space into an appropriate space of EL type, for which the conjecture is already known to hold by the work of Mantovan. [ABSTRACT FROM AUTHOR]

We introduce a new construction of towers of algebraic curves over finite fields and provide a simple example of an optimal tower. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we prove that a projective plane curve defined over a subfield k of C is smooth over k if and only if its associated arcs scheme is reduced. [Copyright &y& Elsevier]

For a cubic differential system having one invariant straight line and a singular point O(0, 0) which is a weak focus, coefficient conditions for O(0, 0) to be a center were found. The results were proved by using the method of rational reversibility. [ABSTRACT FROM AUTHOR]

The article presents a study which demonstrates that pseudoprimes on elliptic curves behave in many ways very similar to pseudoprimes related to Lucas sequences. It mentions that the author derives criteria for the group structure of complex multiplication (CM) curves, which allows for the testing for all composites. Also presented are several formulas that answer questions about certain types of elliptic pseudoprimes.

The parameter space $ \mathcal{S}_p$ is a smooth affine algebraic curve whose genus increases rapidly with $ p$ $ \mathcal{S}_p$ $ \mathcal{S}_p$ $ \mathcal{S}_p$ [ABSTRACT FROM AUTHOR]

We represent and analyze the general solution of the sixth Painlevé transcendent $$ \mathcal{P}_6 $$ in the Picard-Hitchin-Okamoto class in the Painlevé form as the logarithmic derivative of the ratio of τ-functions. We express these functions explicitly in terms of the elliptic Legendre integrals and Jacobi theta functions, for which we write the general differentiation rules. We also establish a relation between the $$ \mathcal{P}_6 $$ equation and the uniformization of algebraic curves and present examples. [ABSTRACT FROM AUTHOR]

The following two questions from univariate (curve) subdivision are not related to each other, but are both of current interest. (1) Is it possible for a non- stationary subdivision scheme to have as its limit an algebraic curve of genus higher than zero? (2) Is it possible to exploit the structure of a (stationary) univariate subdivision matrix to express the unit row eigenvector in closed form? My guess is that both answers are ‘No’, but until proofs of this are articulated it seems worth asking the questions. This note is offered in the hope that a reader will be able to provide some insight. [ABSTRACT FROM AUTHOR]

Abstract: Starting from the well-known transformation law for the Klein functions, we give a proof of a fairly general multiplicative distribution formula for the Siegel functions associated to isogenous complex lattices. This formula has as an immediate consequence the remarkable distribution formula proved by Jarvis in 2000 on the occasion of Rolshausen''s thesis on the second K-group of an elliptic curve. [Copyright &y& Elsevier]

The article provides information on the inequalities of lattice polygons in relation to Scott's inequality and the equation 2i + 7. It states that algebraic geometry is used to study lattice equivalent polygons, since toric geometry is a powerful link that connects discrete and algebraic geometry. Moreover, the proposition b ⩽ 2i + 7 is proven using Pick's Theorem with three equivalent inequalities. Furthermore, Scott's proof indicates that (x,y) = (3,3) is the only point where both upper bounds reach zero and the only case where equality can hold in Onion-Skin Theorem.

We apply the tropical intersection theory as suggested by G. Mikhalkin and developed in detail by L. Allermann and J. Rau to compute intersection products of tropical Psi-classes on the moduli space of rational tropical curves. We show that in the case of zero-dimensional (stable) intersections, the resulting numbers agree with the intersection numbers of Psi-classes on the moduli space of n-marked rational curves computed in algebraic geometry. [ABSTRACT FROM PUBLISHER]

Abstract: In this article we show that the maximum size for the intersection of two distinct ovoids of , q even, is . This is less than the old bound of given by Segre. [Copyright &y& Elsevier]

Abstract: The classical definition of Prym varieties deals with the unramified covers of curves. The aim of this article is to give explicit algebraic descriptions of the Prym varieties associated with ramified double covers of algebraic curves. We make a careful study of the connection with the concept of algebraic completely integrable systems and we apply the methods to some problems such as the Hénon–Heiles system, the Kowalewski rigid body motion and Kirchhoff’s equations of motion of a solid in an ideal fluid. [Copyright &y& Elsevier]

In is proved that, for a regular family of algebraic curves in E n , there exists a sequence of points at which all but the last curvature of the curves simultaneously tend to zero. [ABSTRACT FROM AUTHOR]

In is proved that, for a regular family of algebraic curves in E n , there exists a sequence of points at which all but the last curvature of the curves simultaneously tend to zero. [ABSTRACT FROM AUTHOR]

Let Y → X be an étale cover of regular varieties or arithmetic schemes. We give the following intrinsic characterisation of tameness: Y → X is tamely ramified (outside X), iff for every irreducible curve with normalisation the induced cover Y × X → is tamely ramified (outside ). 2000 Mathematics Subject Classification: 14G40, 14E20. [ABSTRACT FROM AUTHOR]

Abstract: Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for , in particular confirming the parity conjecture for such curves. We prove the analogous results for under the additional assumption that E is not supersingular at primes above 2. [Copyright &y& Elsevier]

Abstract: Complete n-tracks in and non-extendable Near MDS codes of dimension over are known to be equivalent objects. The best known lower bound for the maximum number of points of an n-track is attained by elliptic n-tracks, that is, n-tracks consisting of the -rational points of an elliptic curve. This has given a motivation for the study of complete elliptic n-tracks. From previous work, an elliptic n-track in is complete provided that either the j-invariant of the underlying elliptic curve is different from zero, or and the number of -rational points of is even. In this paper it is shown that the latter result extends to odd if and only if either q is a square or , p being the characteristic of . Some completeness results for elliptic n-tracks in dimensions 3 and 5 are also obtained. [Copyright &y& Elsevier]

Abstract: The order bound gives an in general very good lower bound for the minimum distance of one-point algebraic geometric codes coming from curves. This paper is about a generalization of the order bound to several-point algebraic geometric codes coming from curves. [Copyright &y& Elsevier]

We argue that the existence of genus one fibrations with multisections of high degree on certain Calabi–Yau threefolds implies the existence of pairs of such varieties that are not birational, but are derived equivalent. It also (likely) implies the existence of non-birational counterexamples to the Torelli problem for Calabi–Yau threefolds. [ABSTRACT FROM AUTHOR]

Let W → 2 be the universal Weierstrass family of cubic curves over . For each N ≥ 2, we construct surfaces parameterizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of 2. Since W → 2 is the versal deformation space of a cusp singularity, these surfaces convey information about the level structure on any family of curves of genus g degenerating to a cuspidal curve. Our goal in this note is to determine for which values of N these surfaces are smooth over (0, 0). From a topological perspective, the results determine the homeomorphism type of certain branched covers of S3 with monodromy in SL2 (/N). [ABSTRACT FROM AUTHOR]

A classical theorem says that a planar cubic curve passing through eight of the nine intersection points of two triples of lines passes also through the nineth point (see Theorem 1 for a more precise formulation). Yet another classical theorem has a similar structure (see Theorem 2). One aim of this paper is to reveal the common roots of these theorems and to present a very far generalization for algebraic hypersurfaces of arbitrary degree n in $$\mathbb{R}^{d}$$ with any dimension d. The essential tool to achieve this is the notion of a “skew grid” (see Def. 1). The second aim of the present investigations, is to derive constructions of sets $${\bf S} \subset \mathbb{R}^{d}$$ which are not contained in any algebraic hypersurface. Such sets are well suited as knot sets for multivariate interpolation by polynomial functions in d variables. [ABSTRACT FROM AUTHOR]

The article discusses the mathematical principles and applications concerning the families of plane algebraic curves that contain a special set of points. The case of curves in cages is associated with the elementary methods that presume the familiarity with algebraic geometry and geometrical applications. These applications can be considered as natural generalizations of classical theorems in projective geometry and are associated with familiarity with Pascal's theorem and its comparison with the classical Pascal diagram.

Abstract: We define a Fourier–Mukai transform for a triple consisting of two holomorphic vector bundles over an elliptic curve and a homomorphism between them. We prove that in some cases, the transform preserves the natural stability condition for a triple. We also define a Nahm transform for solutions to natural gauge-theoretic equations on a triple—vortices—and explore some of its basic properties. Our approach combines direct methods with dimensional reduction techniques, relating triples over a curve with vector bundles over the product of the curve with the complex projective line. [Copyright &y& Elsevier]

Abstract: Hwang et al. proposed their generalization of proxy signature schemes based on elliptic curves. However, two attacks are proposed to show that their schemes have serious security flaws. By the first attack, an adversary can forge an illegal proxy signature that verifiers cannot actually find out the original signers of proxy signatures. The second attack is used to change proxy signatures into multi-signatures belonging to the group that actually generates the proxy signatures. To overcome these flaws, our improvement on Hwang et al.’s scheme is also proposed. [Copyright &y& Elsevier]

Let D = p1 p2 · · · p m, where p1, p2, . . . , pm are distinct rational primes with p1 ≡ p2 ≡3(mod 8), p i ≡1(mod 8)(3 ≤ i ≤ m), and m is any positive integer. In this paper, we give a simple combinatorial criterion for the value of the complex L–function of the congruent elliptic curve $$ E_{{D^{2} }} :y^{2} = x^{3} - D^{2} x $$at s = 1, divided by the period ω defined below, to be exactly divisible by 22 m–2, the second lowest 2–power with respect to the number of the Gaussian prime factors of D. As a corollary, we obtain a new series of non–congruent numbers whose prime factors can be arbitrarily many. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton–Dyer. [ABSTRACT FROM AUTHOR]

Abstract The depolarization dyadic of the cubic cavity is examined for off-center positions inside the cavity. It is shown that at an edge two of the diagonal components have ill-defined limiting values and at a corner all three diagonal components are ill defined. Figures illustrating those peculiarities are presented. The explanation is found to be the non-unique limit of the solid angle subtended by an open surface when approaching a point on its rim. [ABSTRACT FROM AUTHOR]

We apply the Murasugi–Tristram inequality to real algebraic curves of odd degree in RP2 with a deep nest, i.e. a nest of the depth k - 1 where 2k + 1 is the degree. For such curves, the ingredients of the Murasugi–Tristram inequality can be computed (or estimated) inductively using the computations for iterated torus links due to Eisenbud and Neumann as the base case of the induction and Conway's skein relation as the induction step. As an example of applications, we prove that some isotopy types are not realizable by M-curves of degree 9. In Appendix B, we give some generalization of the skein relation for Conway polynomial. [ABSTRACT FROM AUTHOR]

We study some geometric configurations related to projections of an irreducible algebraic curve embedded in CP³ onto embedded projective planes. These configurations are motivated by applications to static and dynamic computational vision. More precisely, we study how an irreducible closed algebraic curve X embedded in CP³, of degree d and genus g, can be recovered using its projections from points onto embedded projective planes. The embeddings are unknown. The only input is the defining equation of each projected curve. We show how both the embeddings and the curve in CP³ can be recovered modulo some action of the group of projective transformations of CP³. In particular in the case of two projections, we show how in a generic situation, a characteristic matrix of the pair of embeddings can be recovered. In the process we address dimensional issues and as a result find the minimal number of irreducible algebraic curves required to compute this characteristic matrix up to a finite-fold ambiguity, as a function of their degrees and genus. Then we use this matrix to recover the class of the couple of maps and as a consequence to recover the curve. In a generic situation, two projections define a curve with two irreducible components. One component has degree d(d - 1) and the other has degree d, being the original curve. Then we consider another problem. N projections, with known projection operators and N ≫ 1, are considered as an input and we want to recover the curve. The recovery can be done by linear computations in the dual space and in the Grassmannian of lines in CP³. Those computations are respectively based on the dual variety and on the variety of intersecting lines. In both cases a simple lower bound for the number of necessary projections is given as a function of the degree and the genus. A closely related question is also considered. Each point of a finite closed subset of an irreducible algebraic curve is projected onto a plane from a point: For each point the projection center is different. The projection operators are known. We show when and how the recovery of the algebraic curve is possible, in terms of the degree of the curve, and of the degree of the curve of minimal degree generated by the projection centers. [ABSTRACT FROM AUTHOR]

In Keilen (2003) we gave sufficient conditions for the irreducibility of the family Virr([This symbol cannot be presented in ASCII format]l,...,[This symbol cannot be presented in ASCII format]r) of irreducible curves in the linear system \D\t with precisely r singular points of topological, respectively, analytical, types [This symbol cannot be presented in ASCII format]l,... ,[This symbol cannot be presented in ASCII format]r on several classes of smooth projective surfaces Σ. The conditions were of the form [This symbol cannot be presented in ASCII format] where τ* is some invariant of singularity types, KΣ is the canonical divisor of Σ, and γ is some constant. In the present paper we improve this condition, that is, the constant γ, by a factor of 9. [ABSTRACT FROM AUTHOR]

We will give a new short proof of Bezout's Theorem for complex algebraic curves in P2 which is local. [ABSTRACT FROM AUTHOR]

We show that given a collection of A lines in ℝn, n ≥ 2, the maximum number of their joints (points incident to at least n lines whose directions form a linearly independent set) is O(An/(n-1)). An analogous result for smooth algebraic curves is also proven. [ABSTRACT FROM AUTHOR]

We show that given a collection of A lines in ℝn, n ≥ 2, the maximum number of their joints (points incident to at least n lines whose directions form a linearly independent set) is O(An/(n-1)). An analogous result for smooth algebraic curves is also proven. [ABSTRACT FROM AUTHOR]