Typical convex curves on convex surfaces

In the sense of Baire categories, most convex curves on a smooth twodimensional closed convex surface are smooth. Moreover, if the set of all closed geodesics has empty interior in the space of all convex curves, then most convex curves are strictly convex.This paper was written during the author's visit at Western Washington University, whose substantial support is acknowledged.     

两条Bézier样条的光滑连接

研究了用一条样条曲线把两条不相连接的样条曲线光滑连接起来的问题,给出了连接两条一元n次参数样条曲线为一条新的一元n次参数样条曲线的条件,适用于参数样条曲线添加控制顶点的情形,进一步得到了两条一次、二次、三次Bézier样条曲线在几何连续性下实现自然光滑连接的条件.     

Families of space curves with large cohomology

We investigate space curves with large cohomology. To this end we introduce curves of subextremal type. This class includes all subextremal curves. Based on geometric and numerical characterizations of curves of subextremal type, we show that, if the cohomology is “not too small,” then they can be parameterized by the union of two generically smooth irreducible families; one of them corresponds to the subextremal curves. For curves of negative genus, the general curve of each of these families is also a smooth point of the support of an irreducible component of the Hilbert scheme. The two components have the same (large) dimension and meet in a subscheme of codimension one.     

Smooth curves on a cone which pass through its vertex

We obtain results concerning the existence of smooth curves on the cone over a (possibly singular) plane curve. As an application, these results are used to prove the existence of certain smooth space curves which are the set-theoretic complete intersection of a cone with some other surface.     

Galois Covers of Moduli of Curves

Moduli spaces of pointed curves with some level structure are studied. We prove that for so-called geometric level structures, the levels encountered in the boundary are smooth if the ambient variety is smooth, and in some cases we can describe them explicitly. The smoothness implies that the moduli space of pointed curves (over any field) admits a smooth finite Galois cover. Finally, we prove that some of these moduli spaces are simply connected.     

Quadrature methods for logarithmic-kernel integral equations on closed curves

The recently proposed method of Sloan and Burn for the logarithmic-kernelfirst-kind integral equation on closed curves is modified, soas to permit high-order convergence in appropriate negativenorm even when the solution is not smooth. In addition, theresult of Sloan and Burn for the original method is shown tobe exrtendable from circular to general smooth curves withoutloss.     

Irreducible buchsbaum curves

For every biliaison class C _M of Buchsbaum curves of π ³ we prove that the leftmost shift in which there are smooth and connected curves is the same as for irreducible curves. As a consequence, every irreducible Buchsbaum curve has a flat deformation with cohomology and Hartshorne-Rao module constant which is smooth and connected.     

Smooth curves linked to thick curves

In this paper we construct smooth irreducible spacecurves C which link geometrically by surfaces of minimal degree containingC to curves of generic embedding dimension three. This produces interesting behavior with respect to both C and . The curves link to smooth connected curves by surfaces of low degree butcannot link to smooth connected curves by surfaces of high degree.The curves C give counterexamples to a conjecture of Martin-Deschamps and Perrin.     

Torsion-Free Sheaves and Moduli of Generalized Spin Curves

This article treats compactifications of the space of generalized spin curves. Generalized spin curves, or r-spin curves, are pairs (X,L) with X a smooth curve and L a line bundle whose rth tensor power is isomorphic to the canonical bundle of X. These are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), which have been of interest recently, in part because of their applications to fermionic string theory. Three different compactifications over Z[1/r], all using torsion-free sheaves, are constructed. All three yield algebraic stacks, one of which is shown to have Gorenstein singularities that can be described explicitly, and one of which is smooth. All three compactifications generalize constructions of Deligne and Cornalba done for the case when r=2.     

Smooth Kummer surfaces in projective three-space

In this note we prove the existence of smooth Kummer surfaces in projective three-space containing sixteen mutually disjoint smooth rational curves of any given degree.

     

Bitangents of tropical plane quartic curves

Mathematische Zeitschrift - We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic...     

频谱曲线计算与模拟曲线DFT理论

一、实函数曲线抽样与其光滑模拟曲线的DFT设震动波实函数x(t)之CFT为X(f),x(t)?X(f),     

A Complete Surface in M6 in characteristic <2

We construct in all characteristics p7lt;2 a complete surface in the moduli space of smooth genus 6 curves. The surface is contained in the locus of curves with automorphisms.     

Uniform approximation of curvature for smooth classes of plane curves

We consider problems of approximating the curvature of plane curves from smooth classes by the curvature of elements of smooth finite-dimensional function spaces (trigonometric polynomials, splines with equidistant knots) in the uniform norm.     

The Schubert Triangle Geometry on an Algebraic Surface

For a smooth complex projective surface, and for two families of curves with traditional singularities in it, we enumerate the pairs of curves in each family having two points of contact among them, thus generalizing the double contact formulae known or conjectured by Zeuthen and Schubert in the case of the complex projective plane. The technique we use to this purpose is a particular notion of triangle which can be defined in any smooth surface, thus potentially generalizing to arbitrary surfaces the Schubert technique of triangles.     

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     

Fragmented deformations of primitive multiple curves

A primitive multiple curve is a Cohen-Macaulay irreducible projective curve Y that can be locally embedded in a smooth surface, and such that Y _red is smooth. We study the deformations of Y to curves with smooth irreducible components, when the number of components is maximal (it is then the multiplicity n of Y). We are particularly interested in deformations to n disjoint smooth irreducible components, which are called fragmented deformations. We describe them completely. We give also a characterization of primitive multiple curves having a fragmented deformation.     

Smooth Projective Planes

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to . We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plane curves are smooth collineations, and prove a variety of results analogous to the theory of classical projective planes. *Thanks to Robert Bryant and John Franks.     

Summability of oscillatory integrals over parameters and the boundedness problem for fourier transforms on curves

We find the exact exponent of summability of the Fourier transform of signed measures concentrated on differentiable curves of finite type. We study the behavior of oscillatory integral operators related to the Fourier transform of signed measures concentrated on curves. We obtain necessary and sufficient conditions for the boundedness of the Fourier transform on smooth curves of finite type.