Strong μ\mu-Bases for Rational Tensor Product Surfaces and Extraneous Factors Associated to Bad Base Points and Anomalies at Infinity

We investigate conditions under which the resultant of a $\mu$-basis for a rational tensor product surface is the implicit equation of the surface without any extraneous factors. In this case, we also derive a formula for the implicit degree of the rational surface based only on the bidegree of the rational parametrization and the bidegrees of the elements of the $\mu$-basis without any knowledge of the number or multiplicities of the base points, assuming only that all the base points are local complete intersections. We conclude that in this case the implicit degree of a rational surface of bidegree $(m,n)$ is at most $mn$, so the rational surface must have at least $mn$ base points counting multiplicity. When the resultant of a $\mu$-basis generates extraneous factors, we show how to predict and compute these extraneous factors from either the existence of bad base points or anomalies occurring in the parametrization at infinity. Examples are provided to flesh out the theory

Ruled quartic surfaces, models and classification

New historical aspects of the classification, by Cayley and Cremona, of ruled quartic surfaces and the relation to string models and plaster models are presented. In a ?modern? treatment of the classification of ruled quartic surfaces the classical one is corrected and completed. The string models of Series XIII of some ruled quartic surfaces (manufactured by L. Brill and by M. Schilling) are based on a result of Rohn concerning curves in P1 × P1 of bi-degree (2, 2). This is given here a conceptional proof.The authors thank the referee for his very useful remarks

A survey of the representations of rational ruled surfaces

The rational ruled surface is a typical modeling surface in computer aided geometric design. A rational ruled surface may have different representations with respective advantages and disadvantages. In this paper, the authors revisit the representations of ruled surfaces including the parametric form, algebraic form, homogenous form and Pl¨ucker form. Moreover, the transformations between these representations are proposed such as parametrization for an algebraic form, implicitization for a parametric form, proper reparametrization of an improper one and standardized reparametrization for a general parametrization. Based on these transformation algorithms, one can give a complete interchange graph for the different representations of a rational ruled surface. For rational surfaces given in algebraic form or parametric form not in the standard form of ruled surfaces, the characterization methods are recalled to identify the ruled surfaces from them.Agencia Estatal de Investigació

Exceptional collections, and the Néron–Severi lattice for surfaces

We work out properties of smooth projective varieties X over a (not necessarily algebraically closed) field $\textit{k}$ that admit collections of objects in the bounded derived category of coherent sheaves D$^{b}$(X) that are either full exceptional, or numerically exceptional of maximal length. Our main result gives a necessary and sufficient condition on the Néron–Severi lattice for a smooth projective surface S with χ(O$_{S}$)=1 to admit a numerically exceptional collection of maximal length, consisting of line-bundles. As a consequence we determine exactly which complex surfaces with p$_{g}$=q=0 admit a numerically exceptional collection of maximal length. Another consequence is that a minimal geometrically rational surface with a numerically exceptional collection of maximal length is rational.The author was supported by the Fund for Mathematics at the Institute for Advanced Study and by EPSRC Early Career Fellowship EP/K005545/1

Geometric Information and Rational Parametrization of Nonsingular Cubic Blending Surfaces

The techniques for parametrizing nonsingular cubic surfaces have shown to be of great interest in recent years. This paper is devoted to the rational parametrization of nonsingular cubic blending surfaces. We claim that these nonsingular cubic blending surfaces can be parametrized using the symbolic computation due to their excellent geometric properties. Especially for the specific forms of these surfaces, we conclude that they must be 3, 4, or 5 surfaces, and a criterion is given for deciding their surface types. Besides, using the algorithm proposed by Berry and Patterson in 2001, we obtain the uniform rational parametric representation of these specific forms. It should be emphasized that our results in this paper are invariant under any nonsingular real projective transform. Two explicit examples are presented at the end of this paper

Classification of Simply-Transitive Levi Non-Degenerate Hypersurfaces in C^3

Holomorphically homogeneous Cauchy–Riemann (CR) real hypersurfaces M3⊂C2 were classified by Élie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces M5⊂C3 using a novel Lie algebraic approach independent of any earlier classifications of abstract Lie algebras. Central to our approach is a new coordinate-free formula for the fundamental (complexified) quartic tensor. Our final result has a unique (Levi-indefinite) non-tubular model, for which we demonstrate geometric relations to planar equi-affine geometry

3D Generative Model Latent Disentanglement via Local Eigenprojection

Designing realistic digital humans is extremely complex. Most data-driven generative models used to simplify the creation of their underlying geometric shape do not offer control over the generation of local shape attributes. In this paper, we overcome this limitation by introducing a novel loss function grounded in spectral geometry and applicable to different neural-network-based generative models of 3D head and body meshes. Encouraging the latent variables of mesh variational autoencoders (VAEs) or generative adversarial networks (GANs) to follow the local eigenprojections of identity attributes, we improve latent disentanglement and properly decouple the attribute creation. Experimental results show that our local eigenprojection disentangled (LED) models not only offer improved disentanglement with respect to the state-of-the-art, but also maintain good generation capabilities with training times comparable to the vanilla implementations of the models. Our code and pre-trained models are available at github.com/simofoti/LocalEigenprojDisentangled