Smooth Plane Curves Research Articles

GIT quotient of holomorphic foliations on ℂℙ2 of degree 2 and quartic plane curves

Abstract We study the quotient variety of the space of foliations on ℂ ⁢ ℙ 2 {\mathbb{CP}^{2}} of degree 2 up to change of coordinates. We find the intersection Betti numbers of this variety. As a corollary, we have that these intersection Betti numbers coincide with the intersection Betti numbers of the quotient variety of quartic plane curves. Finally, we give an explicit isomorphism between the space of foliations of degree 2 with different singular points, without invariant lines and the space of smooth quartic plane curves.

Choosing points on cubic plane curves: Rigidity and flexibility

Every smooth cubic plane curve has 9 flex points and 27 sextatic points. We study the following question of Farb: Is it true that the known algebraic structures give all the possible ways to continuously choose n distinct points on every smooth cubic plane curve, for each given positive integer n? We give an affirmative answer to the question when n=9 and 18 (the smallest open cases), and a negative answer for infinitely many n's.

Characterization of tropical plane curves up to genus six

We provide a new forbidden criterion for the realizability of smooth tropical plane curves. This in turn provides us a complete classification of smooth tropical plane curves up to genus six.

Punctual characterization of the unitary flat bundle of weight one PVHS and application to families of curves

In this paper, we consider the problem of pointwise determining the fibres of the flat unitary subbundle of a polarized variation of Hodge structures (PVHS) of weight one. Starting from the associated Higgs field, and assuming the base has dimension 1, we construct a family of (smooth but possibly non-holomorphic) morphisms of vector bundles with the property that the intersection of their kernels at a general point is the fibre of the flat subbundle. We explore the first one of these morphisms in the case of a geometric PVHS arising from a family of smooth projective curves, showing that it acts as the cup-product with some sort of “second-order Kodaira–Spencer class” which we introduce, and check in the case of a family of smooth plane curves that this additional condition is non-trivial.

Smooth plane curves with outer Galois points whose reduced automorphism group is $A_{5}$

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.

Infinitesimal Variation Functions for Families of Smooth Varieties

In this paper we introduce some variation functions associated to the rank of the Infinitesimal Variations of Hodge Structure for a family of smooth projective complex curves. We give some bounds and inequalities and, in particular, we prove that if X is a smooth plane curve, then, there exists a first order deformation xi in H^1(T_X) which deforms X as plane curve and such that xi cdot :H^0(omega _X)rightarrow H^1({mathcal O}_{X}) is an isomorphism. We also generalize the notions of variation functions to higher dimensional case and we analyze the link between IVHS and the weak and strong Lefschetz properties of the Jacobian ring of a smooth hypersurface.

On linear stability and syzygy stability

In previous works, the authors investigated the relationships between linear stability of a generated linear series |V| on a curve C, and slope stability of the syzygy vector bundle M_{V,L} := ker (V otimes mathcal {O}_C rightarrow L). In particular, the second named author and L. Stoppino conjecture that, for a complete linear system |L|, linear (semi)stability is equivalent to slope (semi)stability of M_{L}. The first and third named authors proved that this conjecture holds in the two opposite cases: hyperelliptic and generic curves. In this work we provide a counterexample to this conjecture on any smooth plane curve of degree 7.

Approximating cylinders with bundle-folding plane-symmetric Bricard linkages

This paper proposes a geometric method via isosceles trapezoids or isosceles trapezoidal prisms (ITP) to approximate a smooth planar curve or a cylinder. The method is then extended for designing double-layer cylinder-approximating deployable mechanisms (CDMs) via ITP mechanisms. A bundle-folding plane-symmetric Bricard linkage, which has an isosceles trapezoidal deployed configuration, is developed. The mappings between geometric parameters and the working configurations are discussed. Kinematics of the isosceles trapezoidal Bricard linkage are analyzed. By linking two sets of isosceles trapezoidal Bricard mechanisms and rectangular Bricard mechanisms under certain geometric conditions, the novel ITP mechanism is synthesized. Two types of ITP mechanisms, which have the same folded and deployed configurations but different intermediate configurations, are obtained. The geometric constraints for constructing general CDMs via ITP mechanisms are presented. The construction method is then validated by three cases of approximating a cylindrical cylinder, a parabolic cylinder, and a sinusoidal cylinder, respectively. The proposed method, which can be applied to all cylinders, provides the geometric guide to construct double-layer CDMs and facilitates engineering applications of the plane-symmetric Bricard linkages.

Infinitesimal Darboux transformation and semi-discrete MKDV equation

We connect certain continuous motions of discrete planar curves resulting in semi-discrete potential Korteweg–de Vries (mKdV) equation with Darboux transformations of smooth planar curves. In doing so, we define infinitesimal Darboux transformations that include the aforementioned motions, and also give an alternate geometric interpretation for establishing the semi-discrete potential mKdV equation.

The singular evolutoids set and the extended evolutoids front

In this paper we introduce the notion of the singular evolutoid set which is the set of all singular points of all evolutoids of a fixed smooth planar curve with at most cusp singularities. By the Gauss-Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary (Theorem 2.20 in [4]) applied to the extended front of evolutoids of a hedgehog we obtain an integral equality for smooth periodic curves.

Galois covers of the projective line by smooth plane curves of large degree

Galois covers of the projective line by smooth plane curves of large degree

New examples of Weierstrass semigroups associated with a double covering of a curve on a Hirzebruch surface of degree one

Let \(\varphi :\Sigma _1\longrightarrow {\mathbb {P}}^2\) be a blow up at a point on \({\mathbb {P}}^2\). Let C be the proper transform of a smooth plane curve of degree \(d\ge 4\) by \(\varphi \), and let P be a point on C. Let \(\pi :{\tilde{C}}\longrightarrow C\) be a double covering branched along the reduced divisor on C obtained as the intersection of C and a reduced divisor in \(|-2K_{\Sigma _1}|\) containing P. In this paper, we investigate the Weierstrass semigroup \(H({\tilde{P}})\) at the ramification point \({\tilde{P}}\) of \(\pi \) over P, in the case where the intersection multiplicity at \(\varphi (P)\) of \(\varphi (C)\) and the tangent line at \(\varphi (P)\) of \(\varphi (C)\) is \(d-1\).

Slope equality of plane curve fibrations and its application to Durfee’s conjecture

We give a slope equality for fibered surfaces whose general fiber is a smooth plane curve. As a corollary, we prove a Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee's strong conjecture for such singularities with non-negative topological Euler number of the exceptional set of the minimal resolution.

On the Numerical Approximation of Three-Dimensional Time Fractional Convection-Diffusion Equations

In this paper, we present an efficient method for the numerical investigation of three-dimensional non-integer-order convection-diffusion equation (CDE) based on radial basis functions (RBFs) in localized form and Laplace transform (LT). In our numerical scheme, first we transform the given problem into Laplace space using Laplace transform. Then, the local radial basis function (LRBF) method is employed to approximate the solution of the transformed problem. Finally, we represent the solution as an integral along a smooth curve in the complex left half plane. The integral is then evaluated to high accuracy by a quadrature rule. The Laplace transform is used to avoid the classical time marching procedure. The radial basis functions are important tools for scattered data interpolation and for solving partial differential equations (PDEs) of integer and non-integer order. The LRBF and global radial basis function (GRBF) are used to produce sparse collocation matrices which resolve the issue of the sensitivity of shape parameter and ill conditioning of system matrices and reduce the computational cost. The application of Laplace transformation often leads to the solution in complex plane which cannot be generally inverted. In this work, improved Talbot’s method is utilized which is an efficient method for the numerical inversion of Laplace transform. The stability and convergence of the method are discussed. Two test problems are considered to validate the numerical scheme. The numerical results highlight the efficiency and accuracy of the proposed method.

Tropically planar graphs

We study tropically planar graphs, which are the graphs that appear in smooth tropical plane curves. We develop necessary conditions for graphs to be tropically planar, and compute the number of tropically planar graphs up to genus 7. We provide non-trivial upper and lower bounds on the number of tropically planar graphs, and prove that asymptotically \(0\%\) of connected trivalent planar graphs are tropically planar.

$$K_{-1}$$ and Higher $${\widetilde{K}}$$-Groups of Cones of Smooth Projective Plane Curves

$$K_{-1}$$ and Higher $${\widetilde{K}}$$-Groups of Cones of Smooth Projective Plane Curves

Variations on the Tait–Kneser Theorem

At every point, a smooth plane curve can be approximated, to second order, by a circle; this circle is called osculating. One may think of the osculating circle as passing through three infinitesimally close points of the curve. A vertex of the curve is a point at which the osculating circle hyper-osculates: it approximates the curve to third order. Equivalently, a vertex is a critical point of the curvature function. Consider a (necessarily non-closed) curve, free from vertices. The classical Tait-Kneser theorem [13, 5] (see also [3, 10]), states that the osculating circles of the curve are pairwise disjoint, see Figure 1. This theorem is closely related to the four vertex theorem of S. Mukhopadhyaya [8] that a plane oval has at least 4 vertices (see again [3, 10]). Figure 1 illustrates the Tait-Kneser theorem: it shows an annulus foliated by osculating circles of a curve.

Dihedral Groups and Smooth Plane Curves with Many Quasi-Galois Points

Dihedral Groups and Smooth Plane Curves with Many Quasi-Galois Points

Prism graphs in tropical plane curves

Any smooth tropical plane curve contains a distinguished trivalent graph called its skeleton. In 2020 Morrison and Tewari proved that the so-called big face graphs cannot be the skeleta of tropical curves for genus $12$ and greater. In this paper we answer an open question they posed to extend their result to the prism graphs, proving that they are the skeleton of a smooth tropical plane curve precisely when the genus is at most $11$. Our main tool is a classification of lattice polygons with two points than can simultaneously view all others, without having any one point that can observe all others.

Higher rank Clifford indices of curves on a K3 surface

Let (X, H) be a polarized K3 surface with mathrm {Pic}(X) = mathbb {Z}H, and let Cin |H| be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when gge r^2ge 4, the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree d(ge 5) smooth plane curve is d-4, which is the same as the Clifford index of the curve.