High Degree Curves Research Articles

Sextactic points on the Fermat cubic curve and arrangements of conics

The purpose of this note is to report, in narrative rather than rigorous style, about the nice geometry of 6-division points on the Fermat cubic F and various conics naturally attached to them. Most facts presented here were derived by symbolic algebra programs and the idea of the note is to propose a research direction in the search of conceptual proofs of facts stated here and their generalisations. Extensions in several directions seem possible (taking curves of higher degree and contact to F, studying higher degree curves passing through higher order division points on F, studying curves passing through intersection points of already constructed curves, taking the duals etc.) and we hope some younger colleagues might find pleasure in following proposed paths as well as finding their own.

On the Rank of Quadratic Equations for Curves of High Degree

Let $${\mathcal {C}} \subset {{\mathbb {P}}}^r$$ be a linearly normal curve of arithmetic genus g and degree d. In Saint-Donat (CR Acad Sci Paris Ser A 274: 324–327, 1972), B. Saint-Donat proved that the homogeneous ideal $$I({\mathcal {C}})$$ of $${\mathcal {C}}$$ is generated by quadratic equations of rank at most 4 whenever $$d \ge 2g+2$$ . Also, in Eisenbud et al. (Amer J Math 110: 513–539, 1988) Eisenbud, Koh and Stillman proved that $$I({\mathcal {C}})$$ admits a determinantal presentation if $$d \ge 4g+2$$ . In this paper, we will show that $$I({\mathcal {C}})$$ can be generated by quadratic equations of rank 3 if either $$g=0,1$$ and $$d \ge 2g+2$$ or else $$g \ge 2$$ and $$d \ge 4g+4$$ .

D7 moduli stabilization: the tadpole menace

D7-brane moduli are stabilized by worldvolume fluxes, which contribute to the D3-brane tadpole. We calculate this contribution in the Type IIB limit of F-theory compactifications on Calabi-Yau four-folds with a weak Fano base, and are able to prove a no-go theorem for vast swathes of the landscape of compactifications. When the genus of the curve dual to the D7 worldvolume fluxes is fixed and the number of moduli grows, we find that the D3 charge sourced by the fluxes grows faster than 7/16 of the number of moduli, which supports the Tadpole Conjecture of ref. [1]. Our lower bound for the induced D3 charge decreases when the genus of the curves dual to the stabilizing fluxes increase, and does not allow to rule out a sliver of flux configurations dual to high-genus high-degree curves. However, we argue that most of these fluxes have very high curvature, which is likely to be above the string scale except on extremely large (and experimentally ruled out) compactification manifolds.

Rank 3 quadratic generators of Veronese embeddings

Let $L$ be a very ample line bundle on a projective scheme $X$ defined over an algebraically closed field $\Bbbk$ with ${\rm char}\,\Bbbk \neq 2$. We say that $(X,L)$ satisfies property $\mathsf {QR}(k)$ if the homogeneous ideal of the linearly normal embedding $X \subset {\mathbb {P}} H^{0} (X,L)$ can be generated by quadrics of rank less than or equal to $k$. Many classical varieties, such as Segre–Veronese embeddings, rational normal scrolls and curves of high degree, satisfy property $\mathsf {QR}(4)$. In this paper, we first prove that if ${\rm char}\,\Bbbk \neq 3$ then $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (d))$ satisfies property $\mathsf {QR}(3)$ for all $n \geqslant 1$ and $d \geqslant 2$. We also investigate the asymptotic behavior of property $\mathsf {QR}(3)$ for any projective scheme. Specifically, we prove that (i) if $X \subset {\mathbb {P}} H^{0} (X,L)$ is $m$-regular then $(X,L^{d} )$ satisfies property $\mathsf {QR}(3)$ for all $d \geqslant m$, and (ii) if $A$ is an ample line bundle on $X$ then $(X,A^{d} )$ satisfies property $\mathsf {QR}(3)$ for all sufficiently large even numbers $d$. These results provide affirmative evidence for the expectation that property $\mathsf {QR}(3)$ holds for all sufficiently ample line bundles on $X$, as in the cases of Green and Lazarsfeld's condition $\mathrm {N}_p$ and the Eisenbud–Koh–Stillman determininantal presentation in Eisenbud et al. [Determinantal equations for curves of high degree, Amer. J. Math. 110 (1988), 513–539]. Finally, when ${\rm char}\,\Bbbk = 3$ we prove that $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (2))$ fails to satisfy property $\mathsf {QR}(3)$ for all $n \geqslant 3$.

Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Liénard Systems

In this paper we deal with the Lienard system $$\dot{x}=y, \dot{y}=-f_m(x)y-g_n(x),$$ where $$f_m(x)$$ and $$g_n(x)$$ are real polynomials of degree m and n, respectively. We call this system the Lienard system of type (m, n). For this system, we proved that if $$m+1\le n\le [\frac{4m+2}{3}]$$ , then the maximum number of hyperelliptic limit cycles is $$n-m-1$$ , and this bound is sharp. This result indicates that the Lienard system of type $$(m,m+1)$$ has no hyperelliptic limit cycles. Secondly, we present examples of irreducible algebraic curves of arbitrary high degree for Lienard systems of type $$(m,2m+1)$$ . Moreover, these systems have a rational first integral. Finally, we proved that the Lienard system of type (2, 5) has at most one hyperelliptic limit cycle, and this bound is sharp.

Manin Involutions for Elliptic Pencils and Discrete Integrable Systems

We contribute to the algebraic-geometric study of discrete integrable systems generated by planar birational maps: (a) we find geometric description of Manin involutions for elliptic pencils consisting of curves of higher degree, birationally equivalent to cubic pencils (Halphen pencils of index 1), and (b) we characterize special geometry of base points ensuring that certain compositions of Manin involutions are integrable maps of low degree (quadratic Cremona maps). In particular, we identify some integrable Kahan discretizations as compositions of Manin involutions for elliptic pencils of higher degree.

Designing Developable C-Bézier Surface with Shape Parameters

Developable surface plays an important role in geometric design, architectural design, and manufacturing of material. Bézier curve and surface are the main tools in the modeling of curve and surface. Since polynomial representations can not express conics exactly and have few shape handles, one may want to use rational Bézier curves and surfaces whose weights control the shape. If we vary a weight of rational Bézier curve or surface, then all of the rational basis functions will be changed. The derivation and integration of the rational curve will yield a high degree curve, which means that the shape of rational Bézier curve and surface is not easy to control. To solve this problem of shape controlling for a developable surface, we construct C-Bézier developable surfaces with some parameters using a dual geometric method. This yields properties similar to Bézier surfaces so that it is easy to design. Since C-Bézier basis functions have only two parameters in every basis, we can control the shape of the surface locally. Moreover, we derive the conditions for C-Bézier developable surface interpolating a geodesic.

Braces for idiopathic scoliosis in adolescents.

Braces for idiopathic scoliosis in adolescents.

On the Abelian integrals of quadratic reversible centers with orbits formed by genus one curves of higher degree

In this study, we determine the number of zeros for Abelian integrals in four cases of quadratic reversible centers of genus one. Based on the results of Li et al., we prove that the upper bound of the associated number of zeros for Abelian integrals with orbits formed by quartics, quintics, sextics, and eight-degree curves under polynomial perturbations of arbitrary degree n depends linearly on n.

Hyperkähler Manifolds of Curves in Twistor Spaces

We discuss hypercomplex and hyperk\"ahler structures obtained from higher degree curves in complex spaces fibring over ${\mathbb{P}}^1$.

Representation Method and Degree Selection of Class A Surfaces for Auto-Body

To represent curves and surfaces unified, many commercial soft-wares adopt non-uniform rational B-spline (NURBS) as the method. The NURBS method includes Bézier, B-spline and so forth. In surfaces smoothing Class A surfaces of auto-body are a kind of surfaces that have high desire. While different representing methods can fit curves and surfaces with prodigiously different qualities and efficiency. So the representing methods and degree selection of curves and surfaces for auto-body are fundamental issues that auto-body engineers must be solved. Based on analyzing the character of Bézier and B-spline representing methods, Bézier method is adopted to represent Class A surfaces of auto-body according to the desire of auto-body surfaces. At the same time, the degree of curves and surfaces for auto-body is limited between 2nd-7th according to the oscillation of high degree curves and surfaces and continuity desire. The example verified that the Bézier method can improve the quality and efficiency.

Linear projections and successive minima

LetXbe an arithmetic surface, and letLbe a line bundle onX. Choose a metrichon the lattice Λ of sections ofLoverX. When the degree of the generic fiber ofXis large enough, we get lower bounds for the successive minima of (Λ,h) in terms of the normalized height ofX. The proof uses an effective version (due to C. Voisin) of a theorem of Segre on linear projections and Morrison’s proof that smooth projective curves of high degree are Chow semistable.

Linear projections and successive minima

Let X be an arithmetic surface, and let L be a line bundle on X. Choose a metric h on the lattice Λ of sections of L over X. When the degree of the generic fiber of X is large enough, we get lower bounds for the successive minima of (Λ,h) in terms of the normalized height of X. The proof uses an effective version (due to C. Voisin) of a theorem of Segre on linear projections and Morrison’s proof that smooth projective curves of high degree are Chow semistable.

Modelling of mandibular movement

In this paper, the authors presented a novel method to reconstruct individual mandible movement in three dimensions by a virtual articulator system. The system provides a synchronized 3D mandibular movement that faithfully reappear one's natural occlusion movements—sagittal border, lateral, and open–close movements. Movement trajectories are approximated by cubic or high-degree curves modelling. During the whole process, X-ray exposure on human is avoided. The authors apply the optimal methodology to approach the collected samples of movements in order to generate specific occlusion curves and surfaces. The system provides a novel method of mimicking occlusion of individuals. It is a useful tool for recording dynamic movements of an individual in both pre- and post-surgery and also for predicting surgical outcomes.

A bound on the number of points of a plane curve

A conjecture is formulated for an upper bound on the number of points in PG ( 2 , q ) of a plane curve without linear components, defined over GF ( q ) . We prove a new bound which is half-way from the known bound to the conjectured one. The conjecture is true for curves of low or high degree, or with rational singularity.

Application of Chebyshev II–Bernstein basis transformations to degree reduction of Bézier curves

A polynomial curve on [ 0 , 1 ] can be expressed in terms of Bernstein polynomials and Chebyshev polynomials of the second kind. We derive the transformation matrices that map the Bernstein and Chebyshev coefficients into each other, and examine the stability of this linear map. In the p = 1 and ∞ norms, the condition number of the Chebyshev–Bernstein transformation matrix grows at a significantly slower rate with n than in the power–Bernstein case, and the rate is very close (somewhat faster) to the Legendre–Bernstein case. Using the transformation matrices, we present a method for the best multi-degree reduction with respect to the t − t 2 -weighted square norm for the unconstrained case, which is further developed to provide a good approximation to the best multi-degree reduction with constraints of endpoints continuity of orders r , s ( r , s ≥ 0 ). This method has a quadratic complexity, and may be ill-conditioned when it is applied to the curves of high degree. We estimate the posterior L 1 -error bounds for degree reduction.

Degree reduction of NURBS curves

Higher degree curves are used in applications because they are easier to manipulate interactively but require heavy computation. Most of the equations for curves used popularly in CAD software are of degree 2 and 3, because two curves of degree 3 can guarantee 2nd derivative continuity at the connection point. This study proposes a different but simpler method than any put forward before to deal with degree reduction of free-form curves. The reduced curves use the simplest knot vector type, i.e., the open uniform knot vector. Unlike other methods, this study does not modify or refine the knot vectors but perturb the control points globally. After obtaining an initial condition, a radiating web-like search algorithm is applied to detect the optimum positions. These NURBS curve formats reach basic industrial standards for CAD/CAM/CNC applications. By defining a global bound error function, this algorithm can achieve an optimum solution not only for NURBS curves but also Bezier/B-spline curves.

There is no Bogomolov type restriction theorem for strong semistability in positive characteristic

We give an example of a strongly semistable vector bundle of rank two on the projective plane such that there exist smooth curves of arbitrary high degree with the property that the restriction of the bundle to the curve is not strongly semistable anymore. This shows that a Bogomolov type restriction theorem does not hold for strong semistability in positive characteristic.

Approximation of circular arcs and offset curves by B�zier curves of high degree

In this paper, we present an exact error analysis for circle approximation by Bézier curve. The approximation method is a special case of Floater's conic approximation method (Comput. Aided Geom. Design 8 (1991)135). Using this, we propose an approximation method of the offset curve of given plane Bézier curve by Bézier curve of the same degree.

Approximation of circular arcs and offset curves by Bézier curves of high degree

In this paper, we present an exact error analysis for circle approximation by Bézier curve. The approximation method is a special case of Floater's conic approximation method (Comput. Aided Geom. Design 8 (1991)135). Using this, we propose an approximation method of the offset curve of given plane Bézier curve by Bézier curve of the same degree.