Topic
Cubic surface
About: Cubic surface is a research topic. Over the lifetime, 571 publications have been published within this topic receiving 8026 citations.
Papers published on a yearly basis
1 / 2
Papers
TL;DR: A cubic fourfold is a smooth cubic hypersurface of dimension four; it is special if it contains a surface that is not homologous to a complete intersection as mentioned in this paper.
Abstract: A cubic fourfold is a smooth cubic hypersurface of dimension four; it is special if it contains a surface not homologous to a complete intersection. Special cubic fourfolds form a countably infinite union of irreducible families
$$C_d $$
, each a divisor in the moduli space
$$C $$
of cubic fourfolds. For an infinite number of these families, the Hodge structure on the nonspecial cohomology of the cubic fourfold is essentially the Hodge structure on the primitive cohomology of a K3 surface. We say that this K3 surface is associated to the special cubic fourfold. In these cases,
$$C_d $$
is related to the moduli space
$$N_d $$
of degree d K3 surfaces. In particular,
$$C $$
contains infinitely many moduli spaces of polarized K3 surfaces as closed subvarieties. We can often construct a correspondence of rational curves on the special cubic fourfold parametrized by the K3 surface which induces the isomorphism of Hodge structures. For infinitely many values of d, the Fano variety of lines on the generic cubic fourfold of
$$C_d $$
is isomorphic to the Hilbert scheme of length-two subschemes of an associated K3 surface.
340 citations
Book•
1 Aug 1989TL;DR: A simple, unified, algorithmic approach to change of basis procedures in computer aided geometric design, R.A. Froyland, et al wonderful triangle.
Abstract: Symmetrizing multiaffine polynomials, P.J. Barry norm estimates for inverses of distance matrices, B.J.C. Baxter numerical treatment of surface-surface-intersection and contouring, K.-H. Brakhage modelling closed surfaces - a comparison of existing methods, P. Brunet and A. Vinacua a new characterization of plane elastica, G. Brunnett POLynomials, POLar forms, and interPOL-ation, P.de Casteljau pyramid patches provide potential polynomial paradigms, A.S. Cavaretta and C.A. Micchelli implicitizing rational surfaces with base points by applying perturbations and the factors of zero theorem, E.-W. Chionh and R.C. Goldman wavelets and multiscale interpolation, C.K. Chui and X. Shi a curve intersection algorithm with processing of singular cases - introduction of a clipping technique, M. Daniel best approximation of parametric curves by splines, W.L.F. Degen an approximately G1 cubic surface interpolant, T. DeRose and S. Mann on the G2 continuity of piecewise parametric surfaces, W. Du and F.J.M. Schmitt stationary and non-stationary binary subdivision schemes, N. Dyn and D. Levin MQ-curves are curves in tension, M. Eck offset approximation improvement by control point perturbation, G. Elber and E. Cohen curves and surfaces in geometrical optics, R.T. Farouki and J.-C.A. Chastang evaluation and properties of a derivative of a NURBS curve, M.S. Floater hybrid cubic Bezier triangle patches, T.A. Foley and K. Opitz modelling geological structures using splines, L.A. Froyland, et al wonderful triangle - a simple, unified, algorithmic approach to change of basis procedures in computer aided geometric design, R.N. Goldman an arbitrary mesh network scheme using rational splines, J.A. Gregory and P.K. Yuen Bezier curves and surface patches on quadrics, J. Hoschek monotonicity preserving interplation using C2 rational cubic Bezier curves, M.K. Ismail on piecewise quadratic G2 approximation and interplation, J. Kozak and M. Lokar non-affine blossoms and subdivision for Q-splines, R. Kulkarni. Part contents.
288 citations
144 citations
TL;DR: In this paper, the main result is a boundedness theorem for n-complements on algebraic surfaces, and this theorem is used in a classification of log Del Pezzo surfaces and birational contractions for three-folds.
Abstract: The main result is a boundedness theorem forn-complements on algebraic surfaces. In addition, this theorem is used in a classification of log Del Pezzo surfaces and birational contractions for threefolds.
141 citations
TL;DR: A complete classification of del Pezzo surfaces with quotient singularities and Picard rank 1 which admit a Q-Gorenstein smoothing is given in this article, where the surfaces in each family correspond to solutions of a Markov-type equation and remaining surfaces are obtained as deformations of the toric surfaces or belong to a finite list of sporadic surfaces.
Abstract: We give a complete classification of del Pezzo surfaces with quotient singularities and Picard rank 1 which admit a Q-Gorenstein smoothing
There are 14 infinite families of toric examples The surfaces in each family correspond to solutions of a Markov-type equation The remaining surfaces are obtained as deformations of the toric surfaces or belong to a finite list of sporadic surfaces
129 citations
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| Year | Papers |
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| 2026 | 1 |
| 2025 | 2 |
| 2024 | 2 |
| 2023 | 7 |
| 2022 | 26 |
| 2021 | 22 |