Primitive Curve Research Articles

Primitivoids of curves in Minkowski plane

<abstract><p>In this work, we investigate the differential geometric characteristics of pedal and primitive curves in a Minkowski plane. A primitive is specified by the opposite structure for creating the pedal, and primitivoids are known as comparatives of the primitive of a plane curve. We inspect the relevance between primitivoids and pedals of plane curves that relate with symmetry properties. Furthermore, under the viewpoint of symmetry, we expand these notions to the frontal curves in the Minkowski plane. Then, we present the relationships and properties of the frontal curves in this category. Numerical examples are presented here in support of our main results.</p></abstract>

Virtual χ−y‐genera of Quot schemes on surfaces

This paper studies the virtual $\chi_{-y}$-genera of Grothendieck's Quot schemes on surfaces, thus refining the calculations of the virtual Euler characteristics by Oprea-Pandharipande. We first prove a structural result expressing the equivariant virtual $\chi_{-y}$-genera of Quot schemes universally in terms of the Seiberg-Witten invariants. The formula is simpler for curve classes of Seiberg-Witten length $N$, which are defined in the paper. By way of application, we give complete answers in the following cases: (i) arbitrary surfaces for the zero curve class, (ii) relatively minimal elliptic surfaces for rational multiples of the fiber class, (iii) minimal surfaces of general type with $p_g>0$ for any curve classes. Furthermore, a blow up formula is obtained for curve classes of Seiberg-Witten length $N$. As a result of these calculations, we prove that the generating series of the virtual $\chi_{-y}$-genera are given by rational functions for all surfaces with $p_g>0$, addressing a conjecture of Oprea-Pandharipande. In addition, we study the reduced $\chi_{-y}$-genera for $K3$ surfaces and primitive curve classes with connections to the Kawai-Yoshioka formula.

(Un)distorted stabilisers in the handlebody group

We study geometric properties of stabilisers in the handlebody group. We find that stabilisers of meridians are undistorted, while stabilisers of primitive curves or annuli are exponentially distorted for large enough genus.

Rational curves in holomorphic symplectic varieties and Gromov–Witten invariants

We use Gromov–Witten theory to study rational curves in holomorphic symplectic varieties. We present a numerical criterion for the existence of uniruled divisors swept out by rational curves in the primitive curve class of a very general holomorphic symplectic variety of K3[n] type. We also classify all rational curves in the primitive curve class of the Fano variety of lines in a very general cubic 4-fold, and prove the irreducibility of the corresponding moduli space. Our proofs rely on Gromov–Witten calculations by the first author, and in the Fano case on a geometric construction of Voisin. In the Fano case a second proof via classical geometry is sketched.

DONALDSON–THOMAS INVARIANTS OF LOCAL ELLIPTIC SURFACES VIA THE TOPOLOGICAL VERTEX

We compute the Donaldson–Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in Bryan et al. [‘Trace identities for the topological vertex’, Selecta Math. (N.S.)24 (2) (2018), 1527–1548, arXiv:math/1603.05271], we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the Katz–Klemm–Vafa formula for primitive curve classes which is independent of the computation of Kawai–Yoshioka.

Gromov–Witten Theory of $\text{K3} \times {\mathbb{P}}^1$ and Quasi-Jacobi Forms

Abstract Let $S$ be a K3 surface with primitive curve class $\beta$. We solve the relative Gromov–Witten theory of $S \times {\mathbb{P}}^1$ in classes $(\beta,1)$ and $(\beta,2)$. The generating series are quasi-Jacobi forms and equal to a corresponding series of genus $0$ Gromov–Witten invariants on the Hilbert scheme of points of $S$. This proves a special case of a conjecture of Pandharipande and the author. The new geometric input of the paper is a genus bound for hyperelliptic curves on K3 surfaces proven by Ciliberto and Knutsen. By exploiting various formal properties we find that a key generating series is determined by the very first few coefficients. Let $E$ be an elliptic curve. As collorary of our computations, we prove that Gromov–Witten invariants of $S \times E$ in classes $(\beta,1)$ and $(\beta,2)$ are coefficients of the reciprocal of the Igusa cusp form. We also calculate several linear Hodge integrals on the moduli space of stable maps to a K3 surface and the Gromov–Witten invariants of an abelian threefold in classes of type $(1,1,d)$.

Gromov–Witten invariants of the Hilbert schemes of points of a K3 surface

We study the enumerative geometry of rational curves on the Hilbert schemes of points of a K3 surface. Let $S$ be a K3 surface and let $\mathsf{Hilb}^d(S)$ be the Hilbert scheme of $d$ points of $S$. In case of elliptically fibered K3 surfaces $S \to \mathbb{P}^1$, we calculate genus $0$ Gromov-Witten invariants of $\mathsf{Hilb}^d(S)$, which count rational curves incident to two generic fibers of the induced Lagrangian fibration $\mathsf{Hilb}^d(S) \to \mathbb{P}^d$. The generating series of these invariants is the Fourier expansion of a power of the Jacobi theta function times a modular form, hence of a Jacobi form. We also prove results for genus $0$ Gromov-Witten invariants of $\mathsf{Hilb}^d(S)$ for several other natural incidence conditions. In each case, the generating series is again a Jacobi form. For the proof we evaluate Gromov-Witten invariants of the Hilbert scheme of $2$ points of $\mathbb{P}^1 \times E$, where $E$ is an elliptic curve. Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on $\mathsf{Hilb}^d(S)$ with respect to primitive curve classes. The conjecture is presented in terms of natural operators acting on the Fock space of $S$. We prove the conjecture in the first non-trivial case $\mathsf{Hilb}^2(S)$. As a corollary, we find that the full genus $0$ Gromov-Witten theory of $\mathsf{Hilb}^2(S)$ in primitive classes is governed by Jacobi forms. We present two applications. A conjecture relating genus $1$ invariants of $\mathsf{Hilb}^d(S)$ to the Igusa cusp form was proposed in joint work with R. Pandharipande in \cite{K3xE}. Our results prove the conjecture in case $d=2$. Finally, we present a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through $2$ general points.

Realistic road path reconstruction from GIS data

We introduce a new approach to construct smooth piecewise curves representing realistic road paths. Given a GIS database of road networks in which sampled points are organized in 3D polylines, our method creates horizontal, then vertical curves, and finally combines them to produce 3D road paths. We first estimate the possibility of each point of being a junction between two separate primitive curve segments. Next, we design a tree-traversal algorithm to expand sequences of local best fit primitives which are then merged together with respect to the G1 continuity constraint and civil engineering rules. We apply the Levenberg-Marquardt method to minimize the error between the resulting curve and the sampled points while preserving the G1 continuity.

Local properties of J-complex curves in Lipschitz-continuous structures

We prove the existence of primitive curves and positivity of intersections of J-complex curves for Lipschitz-continuous almost complex structures. These results are deduced from the Comparison Theorem for J-holomorphic maps in Lipschitz structures, previously known for J of class \({\mathcal{C}^{1, Lip}}\). We also give the optimal regularity of curves in Lipschitz structures. It occurs to be \({\mathcal{C}^{1,LnLip}}\), i.e. the first derivatives of a J-complex curve for Lipschitz J are Log-Lipschitz-continuous. A simple example that nothing better can be achieved is given. Further we prove the Genus Formula for J-complex curves and determine their principal Puiseux exponents (all this for Lipschitz-continuous J-s).

Fast Swept Volume Approximation of a General Cutter

A novel algorithm is proposed to fast approximate the swept volume of a general cutter.This algorithm is based on envelope theory and solid modeling techniques.The pipeline of this approximation consists of four stages:1.sampling points on the cutter;2.sweeping the sample points of the cutter along the curved trajectory to create primitive curves;3.creating planar slices along the trajectory,making planar meshes from the intersecting points between slices and primitive curves and approximating envelop curves through the outer boundary vertices of planar meshes;4.making an envelope surface of the swept volume by lofting the envelope curves.Experimental results show that the presented algorithm is simple,efficient and accurate,so it can be widely applied in manufacturing.

Paramétrisation des courbes multiples primitives

Abstract The primitive curves are the multiple curves that can be locally embedded in smooth surfaces (we will always suppose that the associated reduced curves are smooth). These curves have been defined and studied by C. Bănică and O. Forster in 1984. In 1995, D. Bayer and D. Eisenbud gave a complete description of the double curves. We give here a parametrization of primitive curves of arbitrary multiplicity. Let Z n = spec(ℂ[t]/(t n )). The curves of multiplicity n are obtained by taking an open cover (U i ) of a smooth curve C and by glueing schemes of type U i × Z n using automorphisms of U ij × Z n that leave U ij invariant. This leads to the study of the sheaf of nonabelian groups G n of automorphisms of C × Z n that leave the reduced curve invariant, and to the study of its first cohomology set. We prove also that in most cases it is the same to extend a primitive curve C n of multiplicity n to one of multiplicity n + 1, and to extend the quasi locally free sheaf D n of derivations of C n to a rank 2 vector bundle on C n .

Constitutive relations for magnetomechanical hysteresis in ferromagnetic materials

In this work, we present a constitutive relation that appears to give a good description of rate-independent magnetomechanical hysteresis in ferromagnetic materials. Notions such as residual magnetization, magnetic loading (or unloading) function, first magnetization curve, major hysteresis loops, primitive hysteresis loops and derivative curve are introduced. The main arguments which are necessary to construct the magnetic hysteresis loops and the derivative curves are discussed. We also study the effects of couplings with the stress on the overall shape of the magnetic hysteresis loops and the level of saturation. The hysteresis loss associated with a general loop is found.

Function plotting using conic splines

A method is presented whereby, given a mathematical description of a function, a conic spline approximating the plot of the function is produced. Conic arcs were selected as the primitive curves because there are simple incremental plotting algorithms for conics already included in some device drivers, and there are simple algorithms for local approximations by conics. A split-and-merge algorithm for choosing the knots adaptively, according to shape analysis of the original function based on its first-order derivatives, is introduced. >

A diagram to facilitate the study of external ballistics

The semi-graphical method about to do described enables solutions of many dynamical problems of a practical nature to do obtained with ease and rapidity. In particular, useful solutions can be found of problems relating to motion in a resisting medium in cases where the resistance is given as a function of the velocity which cannot be expressed by any simple algebraic form. In this paper the method is applied to the investigation of the motion of a projectile. Some of the curves obtained can be plotted directly from the ballistic table, but none of the curves are found in this way. They are derived by simple graphical processes from a primitive curve which represents the results of numerous experiments, reduced to give the resistance of a standard projectile, fired under standard conditions, as a function of the velocity. The curve, which is the basis of the diagrams given in the paper, is plotted from data given in the official 'Text-book of Small Arms,' 1909. From the educational point of view the method about to be described has the advantage that the essential scientific principles relating to the motion of a projectile can be taught rapidly, and without reference to the laborious methods of analysis by means of which the ballistic tables are derived from the experimental data. As will be seen, other functions, besides those given in the ballistic tables, can be plotted with ease, because elimination is done by mere projection with set-square and T-square. By the drawing of a ballistic diagram a student of gunnery becomes familiar with the general relations existing between the different functions involved; in fact the set of curves constituting the ballistic diagram will probably, when drawn by himself, persist as a permanent mental picture of reference which cannot but be useful to a gunnery officer.

Societies and Academies

Societies and Academies