Low-degree Surface Research Articles

Convex hulls of surfaces in fourspace

AbstractThis is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces exhibit. Our method is a detailed analysis of a general purpose formula by Ranestad and Sturmfels in the case of smooth real algebraic surfaces of low degree (that are rational over the complex numbers). We study both the complex and the real features of the algebraic boundary of Veronese and Del Pezzo surfaces. The main difficulties and the possible approaches to the case of general surfaces are discussed for and complemented by the example of Bordiga surfaces.

Dupin Cyclides Passing through a Fixed Circle

Dupin cyclides are classical algebraic surfaces of low degree. Recently, they have gained popularity in computer-aided geometric design (CAGD) and architecture owing to the fact that they contain many circles. We derive algebraic conditions that fully characterize the Dupin cyclides passing through a fixed circle. The results are applied to the basic problem in CAGD of the blending of Dupin cyclides along circles.

Twistor geometry of the Flag manifold

A study is made of algebraic curves and surfaces in the flag manifold $$\mathbb {F}=SU(3)/T^2$$ , and their configuration relative to the twistor projection $$\pi $$ from $$\mathbb {F}$$ to the complex projective plane $$\mathbb {P}^{2}$$ , defined with the help of an anti-holomorphic involution $$j$$ . This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space $$\mathbb {P}^3$$ of the 4-dimensional sphere $$S^4$$ . Deformations of twistor fibers project to real surfaces in $$\mathbb {P}^{2}$$ , whose metric geometry is investigated. Attention is then focussed on toric del Pezzo surfaces that are the simplest type of surfaces in $$\mathbb {F}$$ of bidegree $$(1,1)$$ . These surfaces define orthogonal complex structures on specified dense open subsets of $$\mathbb {P}^{2}$$ relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree $$(1,1)$$ are determined, and bounds given on the number of twistor fibers that are contained in more general algebraic surfaces in $$\mathbb {F}$$ .

Weak Approximation for del Pezzo Surfaces of Low Degree

Abstract We prove, via an “arithmetic surjectivity” approach inspired by work of Denef, that weak weak approximation holds for surfaces with two conic fibrations satisfying a general assumption. In particular, weak weak approximation holds for general del Pezzo surfaces of degrees $1$ or $2$ with a conic fibration.

Special Ulrich bundles on regular surfaces with non\u2013negative Kodaira dimension

Let S be a regular surface endowed with a very ample line bundle mathcal O_S(h_S). Taking inspiration from a very recent result by D. Faenzi on K3 surfaces, we prove that if {mathcal O}_S(h_S) satisfies a short list of technical conditions, then such a polarized surface supports special Ulrich bundles of rank 2. As applications, we deal with general embeddings of regular surfaces, pluricanonically embedded regular surfaces and some properly elliptic surfaces of low degree in mathbb {P}^{N}. Finally, we also discuss about the size of the families of Ulrich bundles on S and we inspect the existence of special Ulrich bundles on surfaces of low degree.

Curves in $$\mathbb {R}^4$$ and Two-Rich Points

We obtain a new bound on the number of two-rich points spanned by an arrangement of low degree algebraic curves in $\mathbb{R}^4$. Specifically, we show that an arrangement of $n$ algebraic curves determines at most $C_\epsilon n^{4/3+3\epsilon}$ two-rich points, provided at most $n^{2/3+2\epsilon}$ curves lie in any low degree hypersurface and at most $n^{1/3+\epsilon}$ curves lie in any low degree surface. This result follows from a structure theorem about arrangements of curves that determine many two-rich points.

Can GPS-Derived Surface Loading Bridge a GRACE Mission Gap?

We investigated two ‘gap-filler’ methods based on GPS-derived low-degree surface loading variations (GPS-I and GPS-C) and a more simple method (REF-S) which extends a seasonal harmonic variation into the expected Gravity Recovery and Climate Experiment (GRACE) mission gap. We simulated two mission gaps in a reference solution (REF), which is derived from a joint inversion of GRACE (RL05) data, GPS-derived surface loading and simulated ocean bottom pressure. The GPS-I and GPS-C methods both have a new type of constraint applied to mitigate the lack of GPS station network coverage over the ocean. To obtain the GPS-C solution, the GPS-I method is adjusted such that it fits the reference solution better in a 1.5 year overlapping period outside of the gap. As can be expected, the GPS-I and GPS-C solutions contain larger errors compared to the reference solution, which is heavily constrained by GRACE. Within the simulated gaps, the GPS-C solution generally fits the reference solution better compared to the GPS-I method, both in terms of spherical harmonic loading coefficients and in terms of selected basin-averaged hydrological mass variations. Depending on the basin, the RMS-error of the water storage variations (scaled for leakage effects) ranges between 1.6 cm (Yukon) and 15.3 cm (Orinoco). In terms of noise level, the seasonal gap-filler method (REF-S) even outperforms the GPS-I and GPS-C methods, which are still affected by spatial aliasing problems. However, it must be noted that the REF-S method cannot be used beyond the study of simple harmonic seasonal variations.

Splat-based surface reconstruction from defect-laden point sets

We introduce a method for surface reconstruction from point sets that is able to cope with noise and outliers. First, a splat-based representation is computed from the point set. A robust local 3D RANSAC-based procedure is used to filter the point set for outliers, then a local jet surface – a low-degree surface approximation – is fitted to the inliers. Second, we extract the reconstructed surface in the form of a surface triangle mesh through Delaunay refinement. The Delaunay refinement meshing approach requires computing intersections between line segment queries and the surface to be meshed. In the present case, intersection queries are solved from the set of splats through a 1D RANSAC procedure.

Low-degree earth deformation from reprocessed GPS observations

Surface mass variations of low spherical harmonic degree are derived from residual displacements of continuously tracking global positioning system (GPS) sites. Reprocessed GPS observations of 14 years are adjusted to obtain surface load coefficients up to degree nmax = 6 together with station positions and velocities from a rigorous parameter combination. Amplitude and phase estimates of the degree-1 annual variations are partly in good agreement with previously published results, but also show interannual differences of up to 2 mm and about 30 days, respectively. The results of this paper reveal significant impacts from different GPS observation modeling approaches on estimated degree-1 coefficients. We obtain displacements of the center of figure (CF) relative to the center of mass (CM), ΔrCF–CM, that differ by about 10 mm in maximum when compared to those of the commonly used coordinate residual approach. Neglected higher-order ionospheric terms are found to induce artificial seasonal and long-term variations especially for the z-component of ΔrCF–CM. Daily degree-1 estimates are examined in the frequency domain to assess alias contributions from model deficiencies with regard to satellite orbits. Finally, we directly compare our estimated low-degree surface load coefficients with recent results that involve data from the Gravity Recovery and Climate Experiment (GRACE) satellite mission.

On polarized surfaces of low degree whose adjoint bundles are not spanned

Smooth complex surfaces polarized with an ample and globally generated line bundle of degree three and four, such that the adjoint bundle is not globally generated, are considered. Scrolls of a vector bundle over a smooth curve are shown to be the only examples in degree three. Two classes of examples in degree four are presented, one of which is shown to characterize regular such pairs. A Reider-type theorem is obtained in which the assumption on the degree of L is removed.

Surfaces of low degree containing a general canonical curve

Surfaces of low degree containing a general canonical curve

Local bases of G2 continuous G-splines

The theory of G-splines (Höllig and Mögerle, 1990; Mögerle, 1992) has emerged a method based on linear spaces for representing G 2 surfaces over nearly arbitrary rectangular meshes. The only assumption made is that singular vertices, i.e., vertices where n ≠ 4 edges meet, are separated by at least three edges. Our goal is to obtain surfaces of low degree extending the theory of parametric continuity ( C 2), which will be used wherever possible in the mesh. Similar to B-spline basis, basis functions for G-splines are constructed, which are convenient for interpolation. We show their existence and their construction. The linear systems arising from G 2 smoothness constraints in the construction of local bases are quite large. Computer algebra tools to investigate the solvability of these systems are discussed and evaluated.

Smooth curves linked to thick curves

In this paper we construct smooth irreducible space curves which link geometrically by surfaces of minimal degree containing to curves of generic embedding dimension three. This produces interesting behavior with respect to both and . The curves link to smooth connected curves by surfaces of low degree but cannot link to smooth connected curves by surfaces of high degree. The curves give counterexamples to a conjecture of Martin-Deschamps and Perrin.

Algebraic families of smooth hyperbolic surfaces of low degree in ℙ ℂ 3

We reprove (after a paper of Y.T. Siu appeared in 1987) a simple vanishing theorem for the Wronskian of Brody curves under a suitable assumption on the existence of global meromorphic connections. Next we give a slight improvement of a result due to Y.T, Siu and A.M. Nadel (Duke Math. J., 1989) on the algebraic degeneracy of entire holomorphic curves contained in certain hypersurfaces of ℙ ℂ n . Especially, their result is generalized to a larger class of hypersurfaces. Our method produces algebraic families of smooth hyperbolic surfaces in ℙ ℂ 3 for all degreesd≥14; this brings us somewhat nearer than previously known from the expected ranged≥5.

High-order implicit blending surfaces of low degree

The applicability of the so-called potential method for blending implicitly defined surfaces is extended. The extended method is able to produce blendings which are C k -continuous, where k may be chosen arbitrarily large. The blendings consist of piecewise algebraic surfaces of low degree. Specifically, the degree is k + 1 for so-called convex corners, and 2 k for an important class of non-convex corners. In general the degree is linear in k, the constant of proportionality depending on the geometry of the corner. The method is illustrated by a concrete example, where a C 2 blending surface of degree 4 is constructed for a certain non-convex corner.

Analysis of Large-Scale Martian Topography Variations--I: Data Preparation from Earth-Based Radar, Earth-Based CO2 Spectroscopy, and Mariners 6 and 7 CO2 Spectroscopy

Using direct radar ranging of surface heights on Mars and spectrophotometric observations of absorptions produced by carbon dioxide molecules in the Martian atmosphere, data have been obtained on Martian topographical variations at spatial resolutions ranging from about 100 to 1000 km. These data have been studied and analyzed. As a result, a surface height contour map has been produced which clearly reveals a structural complex of blocks and basins whose distribution enhances the magnitude of low-degree surface harmonics. It is emphasized that Mars possesses unexpectedly pronounced topography which can have important geophysical consequences.