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AbstractRational conic fibrations embedded in PN and containing an irreducible curve C transverse to the fibers are classified when C is either a plane curve or a rational normal curve of any degree d. This extends our previous results concerning lines, conics and twisted cubics. To emphasize the role of linear normality, the case in which C is a non-normal rational quartic is also discussed. Moreover, the rank of the d-th jet map of the surface at all points lying on such curves is evaluated. [ABSTRACT FROM AUTHOR]

A method of forming a curved orthogonal coordinate system on a plane and a technique of constructing new surface shapes with curved trapezoidal plans are presented. Multiple examples of curved trapezoidal plans based on different directrix curves and surfaces with the given plans, including combinations of surfaces with different conjugate directrix curves, are illustrated. The proposed technique of surface forming may be used in architecture and construction for development of thin-walled space structures in both urban and industrial buildings. But for the analysis of thin shells, geometric characteristics of the middle surface of the shell are usually used. Vector equation of surfaces with curved trapezoidal plan was used to obtain the formulas for the fundamental form coefficients and surface curvatures. Examples of calculation of the fundamental form coefficients and curvatures of surfaces with particular directrix curves and vertical coordinate functions are presented.

An elementary and self-contained argument for the complete determination of maximal plane curves of degree 3 over F4 will be given, which complements Hirschfeld-Storme-Thas-Voloch’s theorem on a characterization of Hermitian curves in P². This complementary part should be understood as the classification of Sziklai’s example of maximal plane curves of degree q − 1 over Fq. Although two maximal plane curves of degree 3 over F4 up to projective equivalence over F4 appear, they are birationally equivalent over F4 each other. [ABSTRACT FROM AUTHOR]

We introduce and study a new curvature function for plane curves inspired by the weighted mean curvature of M. Gromov. We call it Pólya, being the difference between the usual curvature and the inner product of the normal vector field with the Pólya vector field of a given planar function f. We computed it for several examples, since the general problem of vanishing or constant values of this new curvature involves the general expression of f. [ABSTRACT FROM AUTHOR]

Let X be the normalization of an integral degree d ≤ 9 plane curve Y. We prove that X has a unique gd² if h¹ (P², IZ([d/2]-3)) = 0, where Z is the conductor of Y . Moreover, Y is the unique plane model of X of degree at most d. [ABSTRACT FROM AUTHOR]

There are three well-known Arnold's invariants J + , J − , and St (Arnold strangeness) for plane curves. Two of them, J + and J − , have been successfully quantized. However, the quantization of the Arnold strangeness St remained unrealized. In this paper, we will express a quantization of St by integrating curvatures multiplied by nontrivial densities. The key here is a partition function by Shumakovitch which reformulates the Arnold strangeness. Our quantized Arnold strangeness includes the rotation number, the original Arnold strangeness, and the Tabachnikov invariants as coefficients in its Taylor expansion. [ABSTRACT FROM AUTHOR]

In this paper, some geometric properties of the plane interception curve defined by a nonlinear ordinary differential equation are discussed. Its parametric representation is used to find the limits of some triangle elements associated with the curve. These limits have some connections with the lemniscate constants A , B and Gauss's constant G, which are used to compare with the classical pursuit curve. The analogous spherical geometry problem is solved using a spherical curve defined by the Gudermannian function. It is shown that the results agree with the angle-preserving property of Mercator and Stereographic projections. The Mercator and Stereographic projections also reveal the symmetry of this curve with respect to Spherical and Logarithmic Spirals. The geometric properties of the spherical curve are proved in two ways, analytically and using a lemma about spherical angles. A similar lemma for the planar case is also mentioned. The paper shows symmetry/asymmetry between the spherical and planar cases and the derivation of properties of these curves as limiting cases of some plane and spherical geometry results. [ABSTRACT FROM AUTHOR]

We introduce and study a new curvature function for plane curves inspired by the weighted mean curvature of M. Gromov. We call it rotational being the difference between the usual curvature and the inner product of the normal vector field and the angular vector field. But, since the problem of vanishing of this curvature involves complicated expressions we define a second rotational curvature. Both these curvatures are computed for several examples.

Let C : f = 0 be a reduced curve in the complex projective plane. The minimal degree m d r (f) of a Jacobian syzygy for f , which is the same as the minimal degree of a derivation killing f , is an important invariant of the curve C , for instance it can be used to determine whether C is free or nearly free. In this note we study the relations of this invariant m d r (f) with a decomposition of C as a union of two curves C 1 and C 2 , without common irreducible components. When all the singularities that occur are quasihomogeneous, a result by Schenck, Terao and Yoshinaga yields finer information on this invariant in this setting. Using this, we give some geometrical criteria, the first ones of this type in the existing literature as far as we know , for a line to be a jumping line for the rank 2 vector bundle of logarithmic vector fields along a reduced curve C. [ABSTRACT FROM AUTHOR]

This paper presents a criterion for the existence of a birational embedding into a projective plane with non-collinear Galois points for algebraic curves and describes its application via a novel example of a plane curve with non-collinear Galois points. In addition, this paper presents a new characterisation of the Fermat curve in terms of non-collinear Galois points. [ABSTRACT FROM AUTHOR]

In [8] the first author classified automorphism groups of smooth plane curves of degree not less than four into five types. If the curve has a unique outer Galois point, then the quotient group of its automorphism group by the Galois group at the point, which is called the reduced automorphism group, is a finite subgroup of one-dimensional projective linear group. This article is a sequel of [10] and [9]. In this article, we shall determine the defining equation of the curve when the reduced automorphism group is an icosahedral group and give a description of the full automorphism group. [ABSTRACT FROM AUTHOR]

In [8] the first author classified automorphism groups of smooth plane curves of degree not less than four into five types. If the curve has a unique outer Galois point, then the quotient group of its automorphism group by the Galois group at the point, which is called the reduced automorphism group, is a finite subgroup of one-dimensional projective linear group. This article is a sequel of [10] and [9]. In this article, we shall determine the defining equation of the curve when the reduced automorphism group is an icosahedral group and give a description of the full automorphism group. [ABSTRACT FROM AUTHOR]

A criterion for the existence of a plane model of an algebraic curve such that the Galois closures of projections from two points are the same is presented. Surprisingly, the Hermitian function field in positive characteristic becomes the Galois closure of projections of a plane curve from two non-uniform points. [ABSTRACT FROM AUTHOR]

In this paper, some geometric properties of the plane interception curve defined by a nonlinear ordinary differential equation are discussed. Its parametric representation is used to find the limits of some triangle elements associated with the curve. These limits have some connections with the lemniscate constants A,B and Gauss’s constant G, which are used to compare with the classical pursuit curve. The analogous spherical geometry problem is solved using a spherical curve defined by the Gudermannian function. It is shown that the results agree with the angle-preserving property of Mercator and Stereographic projections. The Mercator and Stereographic projections also reveal the symmetry of this curve with respect to Spherical and Logarithmic Spirals. The geometric properties of the spherical curve are proved in two ways, analytically and using a lemma about spherical angles. A similar lemma for the planar case is also mentioned. The paper shows symmetry/asymmetry between the spherical and planar cases and the derivation of properties of these curves as limiting cases of some plane and spherical geometry results.

In this paper, we investigate a nonlinear evolution equation which is derived from plane curve by Lie symmetry analysis and theory of planar dynamical system. The one-dimensional optimal system of the equation is given and all bifurcations of the system in different parametric regions are given from which we can guarantee the existence of traveling wave solutions including solitary wave solutions, periodic wave solutions and many uncountable infinite traveling wave solutions. Besides, we provide exact expressions of traveling wave solutions under different parametric conditions. [ABSTRACT FROM AUTHOR]

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology. [ABSTRACT FROM AUTHOR]

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This article presents novel examples of plane curves such that the Galois closures of projections from two points are the same, using a criterion described by Fukasawa, Higashine, and Takahashi. [ABSTRACT FROM AUTHOR]

We introduce and study a new curvature function for plane curves inspired by the weighted mean curvature of M. Gromov. We call it generalized radial being the difference between the usual curvature and the inner product of the normal vector field and the gradient of a radial smooth functions. But, since the problem of vanishing of this curvature involves complicated expressions, we computed it for several examples. [ABSTRACT FROM AUTHOR]

First, we give some Weierstrass semigroups which cannot be attained by any smooth curve on a smooth compact toric surface. Next, for any integer l ≥ 2 we describe the Weierstrass semigroup of a total ramification point of a cyclic covering of the projective line with degree l using l − 1 non-negative integers. And finally, we will study smooth curves C lying on a smooth compact toric surface S acted by the torus T which is a dense open subset of S. For an even integer n ≤ 10 we characterize the Weierstrass semigroup of a total ramification point of a cyclic covering of degree n which is the restriction to C of a toric fibration of S such that the ramification point lies on some T -invariant divisor. [ABSTRACT FROM AUTHOR]