Generic determinantal schemes and the smoothability of determinantal schemes of codimension 2

The aim of this article is to prove a criterion for projectively Cohen-Macaulay two-codimensional subschemes ofP _k ^N to be smoothable. For curves inP _k ³ this criterion is due to T. Sauer [4]. The considered schemes are determinantal, so we study related generic affine determinantal schemes. We compute their dimension, and under the condition that the dimension is minimal we calculate the codimension of the singular locus.     

The lazarsfeld-rao property on an arithmetically gorenstein variety

We show how to use all the machinery of liaison techniques for the study of two-codimensional subschemes of a smooth arithmetically Gorenstein subscheme ofP ⁿ.     

Locking constraints for elastic rods and a curvature bound for spatial curves

The paper presents necessary conditions for curves in R³ subject to the nonholonomic constraint of an upper bound for curvature and suitable boundary conditions. The proof essentially uses a reformulation of the problem by means of framed curves. The Euler–Lagrange equations for nonlinearly elastic Cosserat rods subject to a general class of locking constraints is derived by similar methods. Mathematics Subject Classification (2000) 49K15, 51M16, 74B20, 74K10     

Representing a Functional Curve by Curves with Fewer Peaks

In this paper, we study the problems of (approximately) representing a functional curve in 2-D by a set of curves with fewer peaks. Representing a function (or its curve) by certain classes of structurally simpler functions (or their curves) is a basic mathematical problem. Problems of this kind also find applications in applied areas such as intensity-modulated radiation therapy (IMRT). Let f\bf f be an input piecewise linear functional curve of size n. We consider several variations of the problems. (1) Uphill–downhill pair representation (UDPR): Find two nonnegative piecewise linear curves, one nondecreasing (uphill) and one nonincreasing (downhill), such that their sum exactly or approximately represents f\bf f. (2) Unimodal representation (UR): Find a set of unimodal (single-peak) curves such that their sum exactly or approximately represents f\bf f. (3) Fewer-peak representation (FPR): Find a piecewise linear curve with at most k peaks that exactly or approximately represents f\bf f. Furthermore, for each problem, we consider two versions. For the UDPR problem, we study its feasibility version: Given ε>0, determine whether there is a feasible UDPR solution for f\bf f with an approximation error ε; its min-ε version: Compute the minimum approximation error ε ^∗ such that there is a feasible UDPR solution for f\bf f with error ε ^∗. For the UR problem, we study its min-k version: Given ε>0, find a feasible solution with the minimum number k ^∗ of unimodal curves for f\bf f with an error ε; its min-ε version: given k>0, compute the minimum error ε ^∗ such that there is a feasible solution with at most k unimodal curves for f\bf f with error ε ^∗. For the FPR problem, we study its min-k version: Given ε>0, find one feasible curve with the minimum number k ^∗ of peaks for f\bf f with an error ε; its min-ε version: given k≥0, compute the minimum error ε ^∗ such that there is a feasible curve with at most k peaks for f\bf f with error ε ^∗. Little work has been done previously on solving these functional curve representation problems. We solve all the problems (except the UR min-ε version) in optimal O(n) time, and the UR min-ε version in O(n+mlog m) time, where m<n is the number of peaks of f\bf f. Our algorithms are based on new geometric observations and interesting techniques.     

A duality theorem for crossed products of hopf algebras

Abstract

We study the automorphism and collineation groups of plane curves, i.e., in ?², that are not necessarily smooth, and obtain bounds for these curves in terms of, their degree and the number of singularities. We also introduce the notion of bad curves and good curves, and show that all bad curves are rational.     

Recursions for characteristic numbers of genus one plane curves

Characteristic numbers of families of maps of nodal curves toP ² are defined as intersection of natural divisor classes. (This definition agrees with the usual definition for families of plane curves.) Simple recursions for characteristic numbers of genus one plane curves of all degrees are computed.     

On Levels in Arrangements of Curves

Abstract. Analyzing the worst-case complexity of the k -level in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O( nk^ 1-1/(9· 2 ^s-3 ) ) ) for curves that are graphs of polynomial functions of an arbitrary fixed degree s . Previously, nontrivial results were known only for the case s=1 and s=2 . We also improve the earlier bound for pseudo-parabolas (curves that pairwise intersect at most twice) to O( nk ^7/9 log ^2/3 k) . The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudo-parabolas into pseudo-segments, as well as a new observation for cutting pseudo-segments into pieces that can be extended to pseudo-lines. We mention applications to parametric and kinetic minimum spanning trees.     

On Levels in Arrangements of Curves

   Abstract. Analyzing the worst-case complexity of the k -level in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O( nk^ 1-1/(9· 2 ^s-3 ) ) ) for curves that are graphs of polynomial functions of an arbitrary fixed degree s . Previously, nontrivial results were known only for the case s=1 and s=2 . We also improve the earlier bound for pseudo-parabolas (curves that pairwise intersect at most twice) to O( nk ^7/9 log ^2/3 k) . The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudo-parabolas into pseudo-segments, as well as a new observation for cutting pseudo-segments into pieces that can be extended to pseudo-lines. We mention applications to parametric and kinetic minimum spanning trees.     

Remarks on Type III Unprojection

In this note we deal with rational curves in ? ³ which are images of a line by means of a finite sequence of cubo-cubic Cremona transformations. We prove that these curves can always be obtained applying to the line a sequence of such transformations increasing at each step the degree of the curve. As a corollary we get a result about curves that can give speciality for linear systems of ? ³.     

Geometric and cohomological properties of rank 2 vector bundles on curves

Here we study the postulation of curves embedded in a smooth quadric hypersurface of P ⁴ and P ⁵ and relate this subject to the study of cohomological properties of rank 2 spanned vector bundles on smooth projective curves. Received: May 25, 2000; in final form: January 3, 2001?Published online: May 29, 2002     

Convex hulls of generalized moment curves

We consider the family of curves inR ⁴: {fx115-1}, wherep andq are positive integers, and determine the facial structure of the convex hull of these curves.     

Clairaut relation for geodesics of Hopf tubes

Summary In this note we use the Hopf map π: S³ → S² to construct an interesting family of Riemannian metrics h^fon the 3-sphere, which are parametrized on the space of positive smooth functions f on the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. The Hopf tube over an immersed curve γ in S² is the complete lift π^-1(γ) of γ, and we prove that any geodesic of this Hopf tube satisfies a Clairaut relation. A Hopf tube plays the role in S³ of the surfaces of revolution in R^3. Furthermore, we show a geometric integration method of the Frenet equations for curves in those non-standard S³. Finally, if we consider the sphere S³ equipped with a family h^f of Lorentzian metrics, then a new Clairautrelation is also obtained for timelike geodesics of the Lorentzian Hopf tube, and a geometric integration method for curves is still possible.     

A short Proof of Bezout's Theorem in P2

We will give a new short proof of Bezout's Theorem for complex algebraic curves in P ² which is local.     

Osculating Degeneration of Curves

Abstract

The main objects of this paper are osculating spaces of order mto smooth algebraic curves, with the property of meeting the curve again. We prove that the only irreducible curves with an infinite number of this type of osculating spaces of order mare curves in P ^m+1whose degree nis greater than m + 1. This is a generalization of the result and proof of Kaji (Kaji, H. (1986). On the tangentially degenerate curves. J. London Math. Soc.33(2):430–440) that corresponds to the case m = 1. We also obtain an enumerative formula for the number of those osculating spaces to curves in P ^m+2. The case m = 1 of it is a classical formula proved with modern techniques by Le Barz (Le Barz, P. (1982). Formules multisécantes pour les courbes gauches quelconques. In: Enumerative Geometry and Classical Algebraic Geometry. Prog. in Mathematics 24, Birkhäuser, pp. 165–197).     

Rational curves of degree at most 9 on a general quintic threefold

ABSTRACT.

We prove the following form of the Clemens conjecture in low degree. Let d ≤ 9, and let F be a general quintic threefold in P ⁴. Then (1) the Hilbert scheme of rational, smooth and irreducible curves of degree d on F is finite, nonempty, and reduced; moreover, each curve is embedded in F with normal bundle (?1) ⊕ (?1), and in P ⁴ with maximal rank. (2) On F, there are no rational, singular, reduced and irreducible curves of degree d, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3) On F, there are no connected, reduced and reducible curves of degree d with rational components.     

ON THE SEPARATION PROPERTY OF SYMMETRIC ORDINARY SECOND-ORDER DIFFERENTIAL EXPRESSIONS

A 1-variable calculus type argument is used to show that, for a function f : R ² → R, if for all (a, b) ? R ² we have that f o c is smooth for every smooth curve c : R → R ² nonsingular except at 0 and with c(0) = (a, b), then f is smooth. This strengthens Boman's theorem. In fact, we use an even more special collection of smooth curves to prove Boman's theorem. It is shown using a related special collection of smooth curves how the upper half cone can be viewed largely as a model for polar coordinates. Our proof here shows how the use of Frölicher spaces can reduce questions in several dimensions to those of one real variable.     

Level Structures on the Weierstrass Family of Cubics

Let W → 𝔸 ² be the universal Weierstrass family of cubic curves over ?. For each N ≥ 2, we construct surfaces parameterizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of 𝔸 ². Since W → 𝔸 ² is the versal deformation space of a cusp singularity, these surfaces convey information about the level structure on any family of curves of genus g degenerating to a cuspidal curve. Our goal in this note is to determine for which values of N these surfaces are smooth over (0, 0). From a topological perspective, the results determine the homeomorphism type of certain branched covers of S ³ with monodromy in SL₂ (?/N).     

Totally real embeddings of the torus intoC 2

In this short note we combine a construction of Viro and a result of Eliashberg and Harlamov to prove that there exist smooth totally real embeddings of the torus intoC ² which are isotopic but not so within the class of totally real surfaces. We also show how Viro's construction can be used to define an isotopy invariant for a certain class of complex curves inC P ².     

Curves on Generic Surfaces of High Degree Through a Complete Intersection in P 3

We consider general surfaces, S, of high degree containing a given complete intersection space curve, Y. We study integral curves in the subgroup of Pic(S) generated by Y and the plane section. We determine the cohomological invariants of these curves and classify the subcanonical ones. Then using these subcanonical curves we produce stable rank two vector bundles on P ³.