Weddle Surface Research Articles

On Weddle surfaces and their moduli

Abstract The Weddle surface is classically known to be a birational (partially desingularized) model of the Kummer surface. In this note we go through its relations with moduli spaces of abelian varieties and of rank two vector bundles on a genus 2 curve. First we construct a moduli space A 2(3)− parametrizing abelian surfaces with a symmetric theta structure and an odd theta characteristic. Such objects can in fact be seen as Weddle surfaces. We prove that A 2(3)− is rational. Then, given a genus 2 curve C, we give an interpretation of the Weddle surface as a moduli space of extensions classes (invariant with respect to the hyperelliptic involution) of the canonical sheaf ω of C with ω −1. This in turn allows to see the Weddle surface as a hyperplane section of the secant variety Sec(C) of the curve C tricanonically embedded in ℙ4.

Weddle's Surfaces, Humbert's Curves, and a Certain 4-Dimensional Abelian Variety

Weddle's Surfaces, Humbert's Curves, and a Certain 4-Dimensional Abelian Variety

Osculating hyperplanes and a quartic combinant of the nonsingular model of the Kummer and Weddle surfaces

The nonsingular model of Kummer's surface, or its birational equivalent the Weddle surface, is an octavic surface F in [5], projective space of dimension 5 (11), ((10), p. 53), ((1), pp. 218, 219). F is the base variety of a net N of quadrics with a common self-polar simplex S, and has on it 32 lines. These form a single orbit under the group G, of order 32, consisting of the harmonic homologies fixing S.

Non-singular models of specialized Weddle surfaces

There are, in projective space Σ of three dimensions, two famous quartic surfaces: W, the Weddle surface with six nodes ((13), p. 69, footnote) and K, the Kummer surface with sixteen ((s), p. 246, (7) passim). They are in birational correspondence and have the same non-singular model: the octavic base surface F of the net of quadrics in [5], which contain a given line λ and for which a given simplex S is self-polar ((5); (6)). One naturally takes S, with vertices X0, X1, X2, X3, X4, X5 as simplex of reference for homogeneous coordinates x0, x1, x2, x3, x4, x5; F is invariant under the harmonic inversions hj in the vertices Xj and opposite bounding primes xj = 0 of S. These six hj, mutually commutative and having identity for their product, generate an elementary abelian group of order 32. This representation of throws into prominence what may, in this context, be called its positive subgroup , of order 16, consisting of identity and the 15 products hjhk = hkhj; these are harmonic inversions in the edges XjXk and opposite bounding solids xj = xk = 0 of S. The coset of consists of the six hj and their ten products in threes, complementary products being the same (h0h1h2 ≡ h3h4h5) because of the product of all six hj being identity. These ten products are harmonic inversions in the ten pairs of opposite plane faces of S.

Cremona involutions and covariants connected with the Weddle surface

Cremona involutions and covariants connected with the Weddle surface

A Generalization of the Weddle Surface, of Its Cremona Group, and of Its Parametric Expression in Terms of Hyperelliptic Theta Functions

A Generalization of the Weddle Surface, of Its Cremona Group, and of Its Parametric Expression in Terms of Hyperelliptic Theta Functions

On the Geometry of the Weddle Surface

On the Geometry of the Weddle Surface

On the discriminant of ternary forms and a certain class of surfaces

In a paper On the Weddle surface and some analogous locit I have studied the class of surfaces and curves defined as the loci of vertices of general cones passing through certain definite numbers of generic fixed points in space. This investigation may be extended to such systems of cones with nodal generatrices. For this purpose it is obviously necessary to make use of the properties of the discriminant of a ternary form, to which the intersection of a cone with a generic plane reduces.

On the Weddle Surface and Analogous Loci

On the Weddle Surface and Analogous Loci

On the Weddle surface and analogous loci

On the Weddle surface and analogous loci

Complexes of Conics and the Weddle Surface

A complex or system ∞3 of conics in space of four dimensions is such that a finite number of conics pass through an arbitrary point. Linear complexes are those for which this number is unity, and are such that their curves are defined by conditions of incidence with fixed surfaces, curves and points. In this paper are discussed briefly the linear complexes defined by the condition that their curves meet an irreducible curve in four points. Denoting by a curve of order m and genus p it is found that the curves in question are The complex associated with is considered in greater detail, since it is found to have an interesting connection with the well-known Weddle quartic surface of ordinary space. In fact the conics of the system touching a space (of three dimensions) do so in the points of such a surface. The main properties of this surface can be thence deduced. In addition we discuss certain results in connection with this curve . The paper closes with certain enumerative results which were obtained in the course of the researches giving the results recorded and which we believe are worth record.

Relation Between Apolarity and the Weddle Surface

Proceedings of the London Mathematical SocietyVolume s2-21, Issue 1 p. 325-330 Papers Relation Between Apolarity and the Weddle Surface William P. Milne, William P. MilneSearch for more papers by this author William P. Milne, William P. MilneSearch for more papers by this author First published: 1923 https://doi.org/10.1112/plms/s2-21.1.325AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Volumes2-21, Issue11923Pages 325-330 RelatedInformation

An Application of Severi's Theory of a Basis to the Kummer and Weddle Surfaces

An Application of Severi's Theory of a Basis to the Kummer and Weddle Surfaces

An application of Severi’s theory of a basis to the Kummer and Weddle surfaces

An application of Severi’s theory of a basis to the Kummer and Weddle surfaces

Plane Sections of a Weddle Surface

Plane Sections of a Weddle Surface