Nodal Cubics Research Articles

On the geometry of elliptic pairs

An elliptic pair(X,C) is a projective rational surface X with log terminal singularities, and an irreducible curve C contained in the smooth locus of X, with arithmetic genus 1 and self-intersection 0. They are a useful tool for determining whether the pseudo-effective cone of X is polyhedral [1], and interesting algebraic and geometric objects in their own right. Especially of interest are toric elliptic pairs, where X is the blow-up of a projective toric surface at the identity element of the torus. In this paper, we classify all toric elliptic pairs of Picard number two. Strikingly, it turns out that there are only three of these. Furthermore, we study a class of non-toric elliptic pairs coming from the blow-up of P2 at nine points on a nodal cubic, in characteristic p. This construction gives us examples of surfaces where the pseudo-effective cone is non-polyhedral for a set of primes p of positive density, and, assuming the generalized Riemann hypothesis, polyhedral for a set of primes p of positive density.

The nodal cubic and quantum groups at roots of unity

In a recent article, the coordinate ring of the nodal cubic was given the structure of a quantum homogeneous space. Here the corresponding coalgebra Galois extension is expressed in terms of quantum groups at roots of unity, and is shown to be cleft. Furthermore, the minimal quotient extensions are determined.

The Nodal Cubic is a Quantum Homogeneous Space

The cusp was recently shown to admit the structure of a quantum homogeneous space, that is, its coordinate ring B can be embedded as a right coideal subalgebra into a Hopf algebra A such that A is faithfully flat as a B-module. In the present article such a Hopf algebra A is constructed for the coordinate ring B of the nodal cubic, thus further motivating the question which affine varieties are quantum homogeneous spaces.

On the space of orthogonal multiplications in three and four dimensions and Cayley’s nodal cubic

The study of orthogonal multiplications is more than 100 years old and goes back to the works of Hurwitz and Radon. Yet, apart from the extensive literature on admissibility of domain and range dimensions near the Hurwitz-Radon range (in codimension \(\le 8\)), only sporadic and fragmentary results are known about full classification (in large codimension), more specifically, about the moduli space \(\mathfrak {M}_m\) of orthogonal multiplications \(F:\mathbb {R}^m\times \mathbb {R}^m\rightarrow \mathbb {R}^n\) (for various n), even for \(m\le 4\). In this paper we give an insight to the subtle geometries of \(\mathfrak {M}_3\) and \(\mathfrak {M}_4\). Orthogonal multiplications are intimately connected to quadratic eigenmaps between spheres via the Hopf-Whitehead construction. The 9-dimensional moduli space \(\mathfrak {M}_3\) lies on the boundary of the 84-dimensional moduli of quadratic eigenmaps of \(S^5\) into spheres. Similarly, the 36-dimensional moduli space \(\mathfrak {M}_4\) is on the boundary of the 300-dimensional moduli of quadratic eigenmaps of \(S^7\) into spheres. We will show that \(\mathfrak {M}_3\) is the \(SO(3)\times SO(3)\)-orbit of a 3-dimensional convex body bounded by Cayley’s nodal cubic surface with vertices in a real projective space \(\mathbb {R}P^3\), the latter imbedded equivariantly and minimally in an 8-sphere of the space of quadratic spherical harmonics on \(S^3\). For \(\mathfrak {M}_4\), we show that it possesses two orthogonal 18-dimensional slices each of which is an \(SO(4)\times SO(4)\)-orbit of a 6-dimensional polytope \(\mathcal {P}\subset \mathbb {R}^6\). This polytope itself is the convex hull of two orthogonal regular tetrahedra. The corresponding orthogonal multiplications are explicitly constructible. Finally, we give an algebraic description of the 24-dimensional space of diagonalizable elements in \(\mathfrak {M}_4\). The crucial fact in \(\mathbb {R}^4\) is the splitting of the exterior product \(\Lambda ^2(\mathbb {R}^4)\) into self-dual and anti-self-dual components. The techniques employed here can be traced back to the work of Ziller and the author (Toth and Ziller 1999) in describing the 18-dimensional moduli for quartic spherical minimal immersions of \(S^3\) into spheres. As a new feature, we point out the importance of multiplicity of the zeros of the polynomials that define the boundary \(\partial \,\mathfrak {M}_m\) as a determinantal variety.

Interactions between autoequivalences, stability conditions, and moduli problems

We formulate a strong compatibility between autoequivalences and Bridgeland stability conditions and derive a sufficiency criterion. We apply this criterion to an extension of classical slope on the derived category associated to any Galois cover of the nodal cubic. In particular, we give an explicit description of the moduli space of stable objects, its compactification (under S-equivalence), and the group of all autoequivalences compatible with the choice of stability condition.

Computing some fundamental numbers of the variety of nodal cubics in [formula omitted

In this paper we obtain some fundamental numbers of the family U nod of non-degenerate nodal cubics in P 3 involving, in addition to the characteristic conditions, other fundamental conditions, as for example that the node lies on a plane. Some of these numbers were first obtained by Schubert in his Kalkül der abzählenden Geometrie. In our approach we construct several compactifications of U nod , which can be obtained as a sequence of blow-ups of a suitable projective bundle K nod . We also provide geometric interpretations of the degenerations that appear as exceptional divisors. The computations have been carried out with the Wit system.

Computing the characteristic numbers of the variety of nodal plane cubics in [formula omitted

In this note we obtain, phrased in present day geometric and computational frameworks, the characteristic numbers of the family U nod of non-degenerate nodal plane cubics in P 3 , first obtained by Schubert in his Kalkül der abzählenden Geometrie. The main geometric contribution is a detailed study of a variety X nod , which is a compactification of the family U nod , including the boundary components (degenerations) and a generalization to P 3 of a formula of Zeuthen for nodal cubics in P 2 . The computations have been carried out with the Wiris boost WIT.

KP solutions generated from KdV by ‘rank 2’ transference

A sequence of geometric Darboux transformations gives a direct link between any KdV Lax operator and any solution of the KP hierarchy whose associated Burchnall-Chaundy ring is ranked 2 and corresponds to a cuspidal or nodal cubic. This gives a geometric explanation for the Bäcklund transformation that relates KdV to the singular Krichever-Novikov (s-KN) equation. Explicit formulas are developed and applied to the example of the Adler-Moser KdV solutions. to produce s-KN and KP solutions of (fake) rank 2. together with their associated Schur partitions.

Nodal cubic surfaces and the rationality of the moduli space of curves of genus two

Let ~//2 = the moduli space of curves of genus two. It was proved by Igusa in I-2] that ~g2 is a rational variety. Here we propose an easy proof of this fact in the case of a ground field of characteristic zero. Let us give the main step of the proof and an outline of the whole paper. Let S be a "nodal cubic" that is a cubic surface in p3 with a unique ordinary double point P; Ts, e its tangent space at P and TCs, e its tangent cone at P. S c~ T Cs, e is a set of six distinct lines through P. So P(TCs, p) is a conic )7 with six distinguished points P1 . . . . . P6 on it. Let F s be the genus two curve obtained as a double cover of )7 branched at P1 . . . . . P6" Conversely given a genus two curve F its image by the bicanonical map ~p is a conic V C P 2 containing six distinguished points 01 . . . . . 06 which are the branch locus of 02. By blowing up p2 at 01 . . . . . 06 and by blowing down the proper transform of V we obtain a nodal cubic Sr. Then the following facts can be easily checked. (1.1) Any isomorphism ~:$1~$2 , Si nodal cubic in p 3 , i=1 ,2 , is induced by a projective linear transformation in p3.

A Projective Construction for Plane Nodal Cubics

A Projective Construction for Plane Nodal Cubics

On a pencil of nodal cubics. Second paper

On a pencil of nodal cubics. Second paper

Nodal Cubics Through Eight Given Points

Proceedings of the London Mathematical SocietyVolume s2-6, Issue 1 p. 52-57 Papers Nodal Cubics Through Eight Given Points J. E. Wright, J. E. WrightSearch for more papers by this author J. E. Wright, J. E. WrightSearch for more papers by this author First published: 1908 https://doi.org/10.1112/plms/s2-6.1.52Citations: 1AboutPDF 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 Citing Literature Volumes2-6, Issue11908Pages 52-57 RelatedInformation

Proceedings of the London Mathematical Society

THE papers in this volume, as usual, are mostly purely analytical in their character. Prof. Cayley's contributions are very short: the binomial equation xp - I = 0; quinquisection; on the flexure and equilibrium of a skew surface; on the geodesic curvature of a curve on a surface, and on the Gaussian theory of surfaces. Sir J. Cockle continues his remarks on binomial biordinals. Mr. Glaisher's papers are also few and short, viz. on some definite integrals expressible in terms of the first complete definite integral, and of gamma-functions; note on certain symbolic operators and their application to the solution of certain partial differential equations. Messrs. Crofton and J. J. Walker have some points of contact, the former writing on operative symbols in the differential calculus, the latter continuing his theorems in the calculus of operations. Mr. Walker also contributes a quaternion proof of a problem discussed by Mr. S. Roberts, viz. certain tetrahedra specially related to four spheres meeting in a point. Mr. Roberts also gives a historical note on Dr. Graves's theorem on confocal conies.” Mr. W. R. W. Roberts has a paper on the periods of the first class of hyper-elliptic integrals, and a note on the coordinates of a tangent line to the curve of intersection of two quadrics. Mr. T. Craig has a note on Abel's theorem. Papers bearing on geometry are contributed by Prof. Genese, on a system of co-ordinates; by Mr. H. Hart, on the general equation of the second degree in tetrahedral co-ordinates; by Mr. H. M. Jeffery, on bicircular quartics, with a triple and a double focus, and three single foci, all of them colinear; and on spherical quartics, with a quadruple cyclic arc and a triple focus; by Prof. Mannheim, sur les surfaces parallèles; by Mr. R. A. Roberts, on the tangents drawn from a point to a nodal cubic; and note on a system of cartesian ovals, passing through four points on a circle. Signor Brioschi writes sur une propriété du paramètre de la transformée canonique des formes cubiques temaires; and Mr. Carpmael renews an old discussion in his some solutions of Kirkman's 15-school-girl problem. The subject of kinematics on a sphere is ably treated by Mr. E. B. Elliott. Mr. Routh contributes some applications of conjugate functions, and Mr. W. D. Niven writes on the electrical capacity of a conductor bounded by two spherical surfaces cutting at any angle. The presidential address is by Mr. C. W. Merrifield, and is entitled “Considerations respecting the Translation of Series of Observations into Continuous Formulæ.” We have sketched out a bill of fare appealing to many diverse tastes, and we can assure our readers that the dishes are all of admirable quality.