Rational Curves Research Articles

On the local fundamental group of the complement of a curve in a normal surface

We give a presentation of the fundamental group of the complement of a curve $C$ in its "tubular" neighbourhood in a normal surface $S$. The presentation is given in terms of the double weighted dual graph of the resolution of singularities of $C$ (and $S$). This result generalizes the presentation of the fundamental group of the complement of a normal singularity in its neighbourhood given by Mumford in the case, where the dual graph of the resolution is a tree and all exceptional curves of the resolution are rational curves.

The orbit spaces $G_{n,2}/T^n$ and the Chow quotients $G_{n,2}//(\pmb{\mathbb{C}}^{\ast})^n$ of the Grassmann manifolds $G_{n,2}$

The complex Grassmann manifolds $G_{n,k}$ appear as one of the fundamental objects in developing an interaction between algebraic geometry and algebraic topology. The case $k=2$ is of special interest on its own as the manifolds $G_{n,2}$ have several remarkable properties which distinguish them from the $G_{n,k}$ for $k>2$. In our paper we obtain results which, essentially using the specifics of the Grassmann manifolds $G_{n,2}$, develop connections between algebraic geometry and equivariant topology. They are related to well-known problems of the canonical action of the algebraic torus $(\mathbb{C}^{\ast})^n$ on $G_{n,2}$ and the induced action of the compact torus $T^n\subset(\mathbb{C}^{\ast})^n$. Kapranov proved that the Deligne-Mumford-Grothendieck-Knudsen compactification $\overline{\mathcal{M}}(0,n)$ of the space of $n$-pointed rational stable curves can be realized as the Chow quotient $G_{n,2}//(\mathbb{C}^{\ast})^{n}$. In recent papers of the authors a constructive description of the orbit space $G_{n,2}/T^n$ was obtained. In deducing this result the notions of the complex of admissible polytopes and the universal space of parameters $\mathcal{F}_{n}$ for the $T^n$-action on $G_{n,2}$ were of essential use. Using the techniques of wonderful compactification, in this paper an explicit construction of the space $\mathcal{F}_{n}$ is presented. In combination with Keel's description of $\overline{\mathcal{M}}(0,n)$, this construction enabled one to obtain an explicit diffeomorphism between $\mathcal{F}_{n}$ and $\overline{\mathcal{M}}(0,n)$. In this way, we give a description of $G_{n,2}//(\mathbb{C}^{\ast})^{n}$ as the space $\mathcal{F}_{n}$ with a structure described in terms of admissible polytopes $P_\sigma$ and spaces $F_\sigma$. Bibliography: 32 titles.

Символьное вычисление условия резонанса произвольного порядка в системе Гамильтона

The study of formal stability of equilibrium positions of a multiparametric Hamiltonian system in a generic case is traditionally carried out using its normal form under the condition of the absence of resonances of small orders. In this paper we propose a method of symbolic computation of the condition of existence of a resonance of arbitrary order for a system with three degrees of freedom. It is shown that this condition for each resonant vector can be represented as a rational algebraic curve. By methods of computer algebra the rational parametrization of this curve for the case of an arbitrary resonance is obtained. A model example of some two-parameter system of pendulum type is considered.

On the $(n+3)$-webs by rational curves induced by the forgetful maps on the moduli spaces $\mathcal M_{0,n+3}$

On the $(n+3)$-webs by rational curves induced by the forgetful maps on the moduli spaces $\mathcal M_{0,n+3}$

Rank bias for elliptic curves mod p

We conjecture that, for a fixed prime $p$, rational elliptic curves with higher rank tend to have more points mod $p$. We show that there is an analogous bias for modular forms with respect to root numbers, and conjecture that the order of the rank bias for elliptic curves is greater than that of the root number bias for modular forms.

Compactifications of moduli of elliptic K3 surfaces: Stable pair and toroidal

We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal compactifications of the moduli space, one for the ramification divisor and another for the rational curve divisor. In the course of the proof, we further develop the theory of integral affine spheres with 24 singularities. We also construct moduli of rational (generalized) elliptic stable slc surfaces of types ${\bf A_n}$ ($n\ge1$), ${\bf C_n}$ ($n\ge0$) and ${\bf E_n}$ ($n\ge0$).

A characterisation of [formula omitted]-conics of PG(2,q3)

This article considers an Fq-conic contained in an Fq-subplane of PG(2,q3), and shows that it corresponds to a normal rational curve in the Bruck-Bose representation in PG(6,q). The main result characterises which normal rational curves of PG(6,q) correspond via the Bruck-Bose representation to Fq-conics of PG(2,q3). The normal rational curves of interest are called 3-special, a property which describes how the extension of the normal rational curve meets the transversal lines of the regular 2-spread of the Bruck-Bose representation. The proof uses geometric arguments that exploit the interaction between the Bruck-Bose representation of PG(2,q3) in PG(6,q), and the Bose representation of PG(2,q3) in PG(8,q).

An algebraic geometry perspective for the estimation of the directions of arrival

Directions of Arrival for Uniform Linear Arrays represent a widely studied topic in many fields of signal processing, with large attention on computational complexity, estimation accuracy and noise rejection. In this article we study the mathematical model behind Directions of Arrival from the point of view of algebraic geometry, focusing on its relation with the rational normal curve. On this basis, we give a novel interpretation for the root-MUSIC algorithm, that is a widely adopted estimation approach in signal processing. Furthermore, we propose some novel estimation techniques. The first one is based on the computation of the points on the rational normal curve that minimize the distance from the linear subspace defined by the measured Directions of Arrival. The others require the study of the secant varieties of the rational normal curve and the minimization of the distance between the point of the Grassmannian defined by the signal subspace and a certain secant variety. The results obtained from simulations in a noisy scenario show that our estimators are statistically consistent. One algorithm outperforms root-MUSIC over a wide range of scenarios, especially in presence of few snapshots and low Signal to Noise Ratio.

Industrial Robot Contouring Control Based on Non-Uniform Rational B-Spline Curve

This paper presents a novel algorithm about the industrial robot contouring control based on the NURBS (non-uniform rational B-spline) curve. First, aiming at the error between the industrial robot’s actual trajectory and the desired trajectory, the contour error is proposed as the trajectory evaluation index, and the estimation algorithm of contour error based on the tangent approximation is proposed. Based on the tangent approximation algorithm, the estimation algorithm of contour error in the local task coordinate frame is proposed to realize the transformation from the Cartesian coordinate frame to the local task coordinate frame. Second, according to the configuration of the industrial robot, a modified cross-coupling control scheme based on the local task coordinate frame is designed. Finally, the Bernoulli’s lemniscate curves are constructed by NURBS curve and five-order polynomial curve, respectively, and they are symmetrical. The contrast experiment is designed using the two types of constructed Bernoulli’s lemniscate curves as the incentive trajectory. Through the analysis and comparison between the obtained uniaxial tracking error and the contour error curve of the two incentive trajectories, it is concluded that the incentive trajectory constructed by the NURBS curve has better contour control performance than that constructed by the five-order polynomial curve. The results drawn from this paper lay a certain foundation for the future high-precision contouring control of industrial robots.

Curve classes on Calabi–Yau complete intersections in toric varieties

AbstractWe prove the integral Hodge conjecture for curve classes on smooth varieties of dimension at least three constructed as a complete intersection of ample hypersurfaces in a smooth projective toric variety, such that the anticanonical divisor is the restriction of a nef divisor. In particular, this includes the case of smooth anticanonical hypersurfaces in toric Fano varieties. In fact, using results of Casagrande and the toric minimal model program, we prove that in each case, is generated by classes of rational curves.

Superpotentials from singular divisors

We study Euclidean D3-branes wrapping divisors D in Calabi-Yau orientifold compactifications of type IIB string theory. Witten’s counting of fermion zero modes in terms of the cohomology of the structure sheaf {mathcal{O}}_D applies when D is smooth, but we argue that effective divisors of Calabi-Yau threefolds typically have singularities along rational curves. We generalize the counting of fermion zero modes to such singular divisors, in terms of the cohomology of the structure sheaf {mathcal{O}}_{overline{D}} of the normalization overline{D} of D. We establish this by detailing compactifications in which the singularities can be unwound by passing through flop transitions, giving a physical incarnation of the normalization process. Analytically continuing the superpotential through the flops, we find that singular divisors whose normalizations are rigid can contribute to the superpotential: specifically, {h}_{+}^{bullet}left({mathcal{O}}_{overline{D}}right)=left(1,0,0right) and {h}_{-}^{bullet}left({mathcal{O}}_{overline{D}}right)=left(0,0,0right) give a sufficient condition for a contribution. The examples that we present feature infinitely many isomorphic geometric phases, with corresponding infinite-order monodromy groups Γ. We use the action of Γ on effective divisors to determine the exact effective cones, which have infinitely many generators. The resulting nonperturbative superpotentials are Jacobi theta functions, whose modular symmetries suggest the existence of strong-weak coupling dualities involving inversion of divisor volumes.

Explicit classification of isogeny graphs of rational elliptic curves

Let [Formula: see text] be an integer such that [Formula: see text] has genus [Formula: see text], and let [Formula: see text] be a field of characteristic [Formula: see text] or relatively prime to [Formula: see text]. In this paper, we explicitly classify the isogeny graphs of all rational elliptic curves that admit a nontrivial isogeny over [Formula: see text]. We achieve this by introducing [Formula: see text] parameterized families of elliptic curves [Formula: see text] defined over [Formula: see text], which have the following two properties for a fixed [Formula: see text]: the elliptic curves [Formula: see text] are isogenous over [Formula: see text], and there are integers [Formula: see text] and [Formula: see text] such that the [Formula: see text]-invariants of [Formula: see text] and [Formula: see text] are given by the Fricke parameterizations. As a consequence, we show that if [Formula: see text] is an elliptic curve over a number field [Formula: see text] with isogeny class degree divisible by [Formula: see text], then there is a quadratic twist of [Formula: see text] that is semistable at all primes [Formula: see text] of [Formula: see text] such that [Formula: see text].

Subadjunction for Quasi-Log Canonical Pairs and Its Applications

We establish a kind of subadjunction formula for quasi-log canonical pairs. As an application, we prove that a connected projective quasi-log canonical pair whose quasi-log canonical class is anti-ample is simply connected and rationally chain connected. We also supplement the cone theorem for quasi-log canonical pairs. More precisely, we prove that every negative extremal ray is spanned by a rational curve. Finally, we treat the notion of Mori hyperbolicity for quasi-log canonical pairs.

Brill-Noether-general limit root bundles: absence of vector-like exotics in F-theory Standard Models

Root bundles appear prominently in studies of vector-like spectra of 4d F-theory compactifications. Of particular importance to phenomenology are the Quadrillion F-theory Standard Models (F-theory QSMs). In this work, we analyze a superset of the physical root bundles whose cohomologies encode the vector-like spectra for the matter representations (3,2)1/6, ( overline{textbf{3}} ,1)−2/3 and (1,1)1. For the family B3( {Delta }_4^{{}^{circ}} ) consisting of mathcal{O} (1011) F-theory QSM geometries, we argue that more than 99.995% of the roots in this superset have no vector-like exotics. This indicates that absence of vector-like exotics in those representations is a very likely scenario in the mathcal{O} (1011) QSM geometries B3( {Delta }_4^{{}^{circ}} ).The QSM geometries come in families of toric 3-folds B3(∆°) obtained from triangulations of certain 3-dimensional polytopes ∆°. The matter curves in XΣ ∈ B3(∆°) can be deformed to nodal curves which are the same for all spaces in B3(∆°). Therefore, one can probe the vector-like spectra on the entire family B3(∆°) from studies of a few nodal curves. We compute the cohomologies of all limit roots on these nodal curves.In our applications, for the majority of limit roots the cohomologies are determined by line bundle cohomology on rational tree-like curves. For this, we present a computer algorithm. The remaining limit roots, corresponding to circuit-like graphs, are handled by hand. The cohomologies are independent of the relative position of the nodes, except for a few circuits. On these jumping circuits, line bundle cohomologies can jump if nodes are specially aligned. This mirrors classical Brill-Noether jumps. B3( {Delta }_4^{{}^{circ}} ) admits a jumping circuit, but the root bundle constraints pick the canonical bundle and no jump happens.

On the invariant rational curves of a certain family of polynomial differential equations

In this work, we present sufficient conditions to determine if the limit cycles of certain differential systems in the plane are algebraic or not. In particular, we obtain criteria such that the limit cycles of equations derived from predatory prey models with rational functional response are necessarily transcendental ovals.

Curves on K3 surfaces

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic 0, prove almost all cases in positive characteristic, and improve the proofs of the previously known cases. To achieve this, we introduce two new techniques in the deformation theory of curves on K3 surfaces. The first, regeneration, is a process opposite to specialization, which preserves the geometric genus and does not require the class of the curve to extend. The second, called the marked point trick, allows for a controlled degeneration of rational curves to integral ones in certain situations. Combining the two proves existence of integral curves of unbounded degree of any geometric genus g for any projective K3 surface in characteristic 0.

Rational framing motions and spatial rational Pythagorean hodograph curves

We propose a new method for constructing rational spatial Pythagorean Hodograph (PH) curves based on determining a suitable rational framing motion. While the spherical component of the framing motion is arbitrary, the translation part is determined be a modestly sized and nicely structured system of linear equations. Rather surprisingly, generic input data will only result in polynomial PH curves. We provide a complete characterization of all cases that admit truly rational (non-polynomial) solutions. Examples illustrate our ideas and relate them to existing literature.

Relative Stable Pairs and a Non-Calabi–Yau Wall Crossing

Abstract Let $Y$ be a smooth projective threefold, and let $f:Y\to X$ be a birational map with $Rf_*\mathcal {O}_Y=\mathcal {O}_X$. When $Y$ is Calabi–Yau, Bryan–Steinberg (BS) defined enumerative invariants associated to such maps called $f$-relative stable (or BS) invariants. When $X$ has Gorenstein singularities and $f$ has relative dimension one, they compared these invariants to the Donaldson–Thomas, or equivalently Pandharipande–Thomas invariants (PT) of $Y$. We define BS invariants for maps $f$ as above without assuming that $Y$ is Calabi–Yau. For $X$ with Gorenstein and rational singularities, $f$ of relative dimension one, and for insertions from $X$ and arbitrary descendant levels, we conjecture a relation between the generating functions of BS and PT of $Y$. We check the conjecture for the contraction $f: Y\to X$ of a rational curve $C$ with normal bundle $N_{C/Y}\cong \mathcal {O}_C(-1)^{\oplus 2}$ using degeneration and localization techniques to reduce to a Calabi–Yau situation, which we then treat using Joyce’s motivic Hall algebra.

On linear systems with multiple points on a rational normal curve

We give a closed formula for the dimension of all linear systems in Pn with assigned multiplicity at arbitrary collections of points lying on a rational normal curve of degree n. In particular we give a purely geometric explanation of the speciality of these linear systems, which is due to the presence of certain subvarieties in the base locus: linear spans of points, secant varieties of the rational normal curve or joins between them.

Remarks on double points of plane curves

We study the relation between the type of a double point of a plane curve and the curvilinear 0-dimensional subschemes of the curve at the point. An Algorithm related to a classical procedure for the study of double points via osculating curves is described and proved. Eventually we look for a way to create examples of rational plane curves with given singularities A_s.