Nonsingular Curve Research Articles

Unbendable rational curves of Goursat type and Cartan type

We study unbendable rational curves, i.e., nonsingular rational curves in a complex manifold of dimension n with normal bundles isomorphic toOP1(1)⊕p⊕OP1⊕(n−1−p) for some nonnegative integer p. Well-known examples arise from algebraic geometry as general minimal rational curves of uniruled projective manifolds. After describing the relations between the differential geometric properties of the natural distributions on the deformation spaces of unbendable rational curves and the projective geometric properties of their varieties of minimal rational tangents, we concentrate on the case of p=1 and n≤5, which is the simplest nontrivial situation. In this case, the families of unbendable rational curves fall essentially into two classes: Goursat type or Cartan type. Those of Goursat type arise from ordinary differential equations and those of Cartan type have special features related to contact geometry. We show that the family of lines on any nonsingular cubic 4-fold is of Goursat type, whereas the family of lines on a general quartic 5-fold is of Cartan type, in the proof of which the projective geometry of varieties of minimal rational tangents plays a key role.

Anonymous Authentication and Key Agreement Scheme Combining the Group Key for Vehicular Ad Hoc Networks

Vehicular ad hoc network (VANET) is a multihop mobile wireless communication network that can realize many vehicle-related applications through multitop communication. In the open wireless communication environment, security and privacy protection are important contents of VANET research. The most basic method of VANET privacy protection is anonymous authentication. Even through, there are many existing schemes to provide anonymous authentication for VANETs. Many existing schemes suffer from high computational cost by using bilinear pairing operation or need the assistance of the trust authorities (TAs) during the authentication process or rely on an ideal tamper-proof device (TPD), which requires very strong security assumption. In this study, an anonymous authentication and key negotiation scheme by using private key and group key is proposed, which is based on pseudonym using the nonsingular elliptic curve. In this scheme, there is no third party trust center to participate in the authentication, there is no need to query the database, and there is no need of the local database to save the identity information of many vehicles, which reduce the storage space and the authentication time compared with other schemes. The proposed scheme only needs realistic TPDs. In the proposed scheme, TPDs do not need to preinstall the system key as many other schemes do; hence, the failure of a single TPD does not affect the security of the entire system. The security of the scheme is proved under the random oracle model. Compared with the related schemes using bilinear pairings, the computational cost and communication cost of the proposed scheme are reduced by 82% and 50%, respectively.

The Hurwitz curve over a finite field and its Weierstrass points for the morphism of lines

For any smooth Hurwitz curve Hn:XYn+YZn+XnZ=0 over the finite field Fp, an explicit description of its Weierstrass points for the morphism of lines is presented. As a consequence, bounds on the number of rational points of Hn are obtained via Stöhr-Voloch Theory. Further, the full automorphism group Aut(Hn), as well as the genera of all Galois subcovers of Hn, with n≠3,pr, are computed. Finally, a question by F. Torres on plane nonsingular maximal curves is answered.

On Abel-Jacobi maps of moduli of parabolic bundles over a curve

The Abel-Jacobi maps generalize the usual Jacobi map for curves to the case of nonsingular complex projective varieties. Here, we study certain Abel-Jacobi maps on the moduli space of parabolic stable bundles of fixed rank, determinant and generic weight over a nonsingular complex projective curve. We prove that the maps are split surjections.

On planar embeddings of pseudoplanar affine curves of genus ≤1

We show that any two planar cubic embeddings of a pseudoplanar curve of genus 1 (over complex numbers) are linearly equivalent. We describe the ‘abstract moduli’ of nonsingular affine plane cubic curves with 3 places at infinity. We note some examples of inequivalent planar embeddings of a nonsingular affine curve of genus 0 having 2 places at infinity.

Real algebraic curves of bidegree (5,5) on the quadric ellipsoid

We complete the topological classification of real algebraic non-singular curves of bidegree $(5, 5)$ on the quadric ellipsoid. We show in particular that previously known restrictions form a complete system for this bidegree. Therefore, the main part of the paper concerns the construction of real algebraic curves. Our strategy is first to reduce to the construction of curves in the second Hirzebruch surface by degenerating the quadric ellipsoid to the quadratic cone. Next, we combine different classical construction methods on toric surfaces, such as Dessin d'enfants and and Viro's patchworking method.

Cubic Curves Over the Finite Field of Order Twenty-Five

An arc of degree three is a set of points in projective plane no four of which are collinear but some three are collinear, and a cubic curve is a non-singular projective plane cubic curve. There are cubic curves formed an arc of degree three over a finite field. The aims of this paper are to give the inequivalent cubic curves forms over the finite field of order twenty-five according to its inflexion points, and the incomplete curves have been extended to complete arcs of degree three. As a conclusion over F 25, the largest arc size of degree three constructed from the points of cubic curves is 36; that is, 36≤ m r (2,25) ≤51.

Segre invariant and a stratification of the moduli space of coherent systems

The aim of this paper is to generalize the [Formula: see text]-Segre invariant for vector bundles to coherent systems. Let [Formula: see text] be a non-singular irreducible complex projective curve of genus [Formula: see text] and [Formula: see text] be the moduli space of [Formula: see text]-stable coherent systems of type [Formula: see text] on [Formula: see text]. For any pair of integers [Formula: see text] with [Formula: see text], [Formula: see text] we define the [Formula: see text]-Segre invariant, and prove that it defines a lower semicontinuous function on the families of coherent systems. Thus, the [Formula: see text]-Segre invariant induces a stratification of the moduli space [Formula: see text] into locally closed subvarieties [Formula: see text] according to the value [Formula: see text] of the function. We determine an above bound for the [Formula: see text]-Segre invariant and compute a bound for the dimension of the different strata [Formula: see text]. Moreover, we give some conditions under which the different strata are nonempty. To prove the above results, we introduce the notion of coherent systems of subtype [Formula: see text].

Cubic Curves Over the Finite Field of Order Twenty Seven

A cubic curve is a non-singular projective plane cubic curve. An (k; 3)-arc is a set of points no four are collinear but some three are linear. Most of the cubic curve can be regarded as an arc of degree three. In this paper, the projectively inequivalent cubic curves have been classified over the finite field of order twenty-seven with respect to its inflexions points and determined if they are complete or incomplete as arcs of degree three. Also the size of the largest arc of degree three that can be constructed form each incomplete cubic curve are given. The main conclusion that can be drawn is that, over F 27, the largest an arc of degree three can be constructed depending on the cubic curve is 38; that is, 38 ≤ m 3(2,27) ≤ 55.

The existence of supersingular curves of genus 4 in arbitrary characteristic

We prove that there exists a supersingular nonsingular curve of genus 4 in arbitrary characteristic p>0. For p>3 we shall prove that the desingularization of a certain fiber product over mathbf{P }^1 of two supersingular elliptic curves is supersingular.

Zero cycles on the moduli space of curves

While the Chow groups of 0-dimensional cycles on the moduli spaces of Deligne-Mumford stable pointed curves can be very complicated, the span of the 0-dimensional tautological cycles is always of rank 1. The question of whether a given moduli point [C,p_1,...,p_n] determines a tautological 0-cycle is subtle. Our main results address the question for curves on rational and K3 surfaces. If C is a nonsingular curve on a nonsingular rational surface of positive degree with respect to the anticanonical class, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the virtual dimension in Gromov-Witten theory of the moduli space of stable maps. If C is a nonsingular curve on a K3 surface, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the genus of C and every marking is a Beauville-Voisin point. The latter result provides a connection between the rank 1 tautological 0-cycles on the moduli of curves and the rank 1 tautological 0-cycles on K3 surfaces. Several further results related to tautological 0-cycles on the moduli spaces of curves are proven. Many open questions concerning the moduli points of curves on other surfaces (Abelian, Enriques, general type) are discussed.

Global Prym–Torelli theorem for double coverings of elliptic curves

The Prym variety for a branched double covering of a nonsingular projective curve is defined as a polarized abelian variety. We prove that any double covering of an elliptic curve which has more than $4$ branch points is recovered from its Prym variety.

Holomorphic differentials of generalized Fermat curves

A non-singular complete irreducible algebraic curve Fk,n, defined over an algebraically closed field K, is called a generalized Fermat curve of type (k,n), where n,k≥2 are integers and k is relatively prime to the characteristic p of K, if it admits a group H≅Zkn of automorphisms such that Fk,n/H is isomorphic to PK1 and it has exactly (n+1) cone points, each one of order k. By the Riemann-Hurwitz-Hasse formula, Fk,n has genus at least one if and only if (k−1)(n−1)>1. In such a situation, we construct a basis, called a standard basis, of its space H1,0(Fk,n) of regular forms, containing a subset of cardinality n+1 that provides an embedding of Fk,n into PKn whose image is the fiber product of (n−1) classical Fermat curves of degree k. For p=2, we obtain a lower bound (which is sharp for n=2,3) for the dimension of the space of exact one-forms, that is, the kernel of the Cartier operator. We also do this for (p,k,n)=(3,2,4).

Singularities and syzygies of secant varieties of nonsingular projective curves

In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus $g$ is greater than $2g+2k+p$ for nonnegative integers $k$ and $p$, then the $k$-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen--Macaulay, and satisfies the property $N_{k+2, p}$. In addition, the singularities of the secant varieties are further classified according to the genus of the curve, and the Castelnuovo--Mumford regularities are also obtained as well. As one of the main technical ingredients, we establish a vanishing theorem on the Cartesian products of the curve, which may have independent interests and may find applications elsewhere.

Аналитические и теоретико-числовые свойства двумерных сигма-функций

Обзор посвящен классическим и современным задачам, связанным с целой функцией $\sigma({\bf u};\lambda)$, которая определяется семейством неособых алгебраических кривых рода 2, где ${\bf u}= (u_1,u_3)$, $\lambda=(\lambda_4, \lambda_6, \lambda_8, \lambda_{10})$. Эта функция является аналогом сигма-функции Вейерштрасса $\sigma({{u}};g_2,g_3)$ семейства эллиптических кривых. Логарифмические производные порядка 2 и выше функции ${\sigma({\mathbf{u}};\lambda)}$ порождают поле гиперэллиптических функций от ${\mathbf{u}} = (u_1,u_3)$ на якобианах кривых с фиксированным значением вектора параметров $\lambda$. Мы рассматриваем три ряда Гурвица $\sigma({\mathbf{u}};\lambda)=\sum_{m,n\ge0}a_{m,n}(\lambda)\frac{u_1^mu_3^n}{m!n!}$, $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\xi_k(u_1;\lambda)\frac{u_3^k}{k!}$ и $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\mu_k(u_3;\lambda)\frac{u_1^k}{k!}$. Обзор посвящен теоретико-числовым свойствам функций $a_{m,n}(\lambda)$, $\xi_k(u_1;\lambda)$ и $\mu_k(u_3;\lambda)$. Он включает самые последние результаты, доказательства которых использует тот фундаментальный факт, что функция $\sigma ({\mathbf{u}};\lambda)$ определяется системой четырех уравнений теплопроводности в неголономном репере шестимерного пространства.

On the disposition of cubic and pair of conics in a real projective plane

In the first part of the 16th Hilbert problem the question about the topology of nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of algebraic manifolds with singularities belong to this subject too. The particular case of this problem is the study of curves that are decompozable into the product of curves in a general position. This paper deals with the problem of topological classification of mutual positions of a nonsingular curve of degree three and two nonsingular curves of degree two in the real projective plane. Additiolal conditions for this problem include general position of the curves and its maximality; in particular, the number of common points for each pair of curves-factors reaches its maximum. It is proved that the classification contains no more than six specific types of positions of the species under study. Four position types are built, and the question of realizability of the two remaining ones is open.

Elliptic Curves in Hyper-Kähler Varieties

Abstract We show that the moduli space of elliptic curves of minimal degree in a general Fano variety of lines of a cubic four-fold is a non-singular curve of genus $631$. The curve admits a natural involution with connected quotient. We find that the general Fano contains precisely $3,780$ elliptic curves of minimal degree with fixed (general) $j$-invariant. More generally, we express (modulo a transversality result) the enumerative count of elliptic curves of minimal degree in hyper-Kähler varieties with fixed $j$-invariant in terms of Gromov–Witten invariants. In $K3^{[2]}$-type this leads to explicit formulas of these counts in terms of modular forms.

On certain type of singular varieties with smooth subvarieties

Let k an algebraically closed field, $$char\ k=0.$$ Let C be an irreducible nonsingular curve such that $$kC=S\cap X$$, $$k\in {\mathbb {N}},$$ where S and X are two surfaces and all the singularities of X are of the form $$z^{p}=x^{s}-y^{s},$$$$p, \, s$$ primes, $$s=pt+m,$$$$t\in {\mathbb {N}}$$, $$m=1$$ or 2. We also study the cases, for $$p, \, s$$ primes, $$p=4r+1,$$$$r \in {\mathbb {N}},$$$$s=pt+2r+1,$$$$t \in {\mathbb {N}},$$ and $$p=4r+3,$$$$r \in {\mathbb {N}},$$$$s=pt+2r+2,$$$$t \in {\mathbb {N}}.$$ We prove that C can never pass through such kind of singularities of a surface, unless $$k=pa,$$$$a\in {\mathbb {N}}$$. We study multiplicity-k structures on varieties, $$k\in {\mathbb {N}}.$$ Let Z be a reduced irreducible nonsingular $$(n-1)$$-dimensional variety such that $$kZ=X\cap S$$, where S is a $$(N-1)$$-fold in $${\mathbb {P}}^{N},$$X is a normal n-fold with certain type of singularities, like linear compound $$V_{ps}$$ singularity or (d,l) complete intersection compound $$V_{ps}$$ singularity, We study when $$Z\cap \text {Sing} (X)\ne \emptyset .$$ These results generalize some results in Gonzalez-Dorrego (On singular varieties with smooth subvarieties, singularities in geometry, topology, foliations and dynamics. Birkhauser, Basel, pp 125–134, 2017). Seifert invariants of $$z^{p}=x^{s}-y^{s},$$$$p, \, s$$ primes, $$s=pt+m,$$$$t\in {\mathbb {N}}$$, $$m=1,$$ 2 and for $$p=4r+1,$$$$r \in {\mathbb {N}},$$$$s=pt+2r+1,$$$$t \in {\mathbb {N}},$$ and$$p=4r+3,$$$$r \in {\mathbb {N}},$$$$s=pt+2r+2,$$$$t \in {\mathbb {N}}.$$ are studied (Prop. 34).

Results on the Projective Plane over a Finite Field of Order Seventeen

The main goal of this research is to find the projective mapping that transforms a geometric formation called an i -set onto an arc such that the domain of the mapping is a subset of the projective line PG (1,q), q=17 , such that a5-set is called a pentad, a6-set is a hexad, a7-set is a heptad, a8-set is an octad, and a9 -set is a nonad, mapped onto a conicY2-XZ. The research also aims to find the stabilizer group of points on a non-singular cubic curve, with or without rational inflection points, on the projective plane over a finite field of order seventeen, and to give some examples.

On the growth behaviour of Hironaka quotients

Any ruled surface in Euclidean 3-space is described as a curve of unit dual vectors in the algebra of dual quaternions (=the even Clifford algebra of type (0,3,1)). Combining this classical framework and Singularity Theory, we characterize local diffeomorphic types of singular ruled surfaces in terms of geometric invariants. In particular, using a theorem of G. Ishikawa, we show that local topological type of singular (non-cylindrical) developable surfaces is completely determined by vanishing order of the dual torsion, that generalizes an old result of D. Mond for tangent developables of non-singular space curves. Our approach would be useful for analysis on singularities arising in differential line geometry related with several applications such as robotics, vision theory and architectural geometry.