Birationally rigid singular double quadrics and double cubics

Mathematical Notes - In this paper it is shown that Fano double quadrics of index 1 and dimension at least 6 are birationally superrigid if the branch divisor has at most quadratic singularities of...     

Duality and Riemannian cubics

Riemannian cubics are curves used for interpolation in Riemannian manifolds. Applications in trajectory planning for rigid bodiy motion emphasise the group SO(3) of rotations of Euclidean 3-space. It is known that a Riemannian cubic in a Lie group G with bi-invariant Riemannian metric defines a Lie quadratic V in the Lie algebra, and satisfies a linking equation. Results of the present paper include explicit solutions of the linking equation by quadrature in terms of the Lie quadratic, when G is SO(3) or SO(1,2). In some cases we are able to give examples where the Lie quadratic is also given in closed form. A basic tool for constructing solutions is a new duality theorem. Duality is also used to study asymptotics of differential equations of the form , where β₀,β₁ are skew-symmetric 3×3 matrices, and x :ℝ→ SO(3). This is done by showing that the dual of β₀+tβ₁ is a null Lie quadratic. Then results on asymptotics of x follow from known properties of null Lie quadratics. To Charles Micchelli, with warm greetings and deep respect, on his 60th birthday Mathematics subject classifications (2000) 53A17, 53B20, 65D18, 68U05, 70E60.     

Uniqueness of exceptional singular quartics

We prove that given a general collection of 14 points of ( an infinite field) there is a unique quartic hypersurface that is singular on .

This completes the solution to the open problem of the dimension of a linear system of hypersurfaces of that are singular on a collection of general points.

     

Solving cubics by polynomial fitting

A top-performance algorithm for solving cubic equations is introduced. This algorithm uses polynomial fitting for a decomposition of the given cubic into a product of a quadratic and a linear factor. This factorization can be computed extremely accurately and efficiently using a fixed-point iteration of the linearized fitting error. The polynomial fitting concept performs orders of magnitude better in terms of numerical accuracy and precision than any of the currently known and available algorithms for solving cubic equations. A special exception handler is presented for a reliable operation in the event of double, triple and tightly clustered roots.     

All cubics are self-isogonal

We develop techniques to prove that a cubic curve is invariant under the isogonal transformation of the projective plane determined by some triangle.     

Twisted cubics, bis

We consider the smooth compactification constructed in [12] for a space of varieties like twisted cubics. We show this compactification embeds naturally in a product of flag varieties.Partially supported by CNPq, Pronex (ALGA)     

Fano surfaces of real quartics

We prove certain properties of the Fano surface of bitangents to the real quartic in projective 3-space. In particular, we estimate the dimension of the cohomology of the real part of this surface in terms of the dimension of the cohomology of the real part of the quartic, and compute its Euler characteristic.     

On the Jacobians of plane cubics

Let S be a scheme and f a ternary cubic form whose ten coefficients are sections of O_S without common zero. The equation f=0 defines a family of plane cubic curves parametrized by S. We prove that the family of generalized Jacobians of those cubic curves is a group scheme J/S which is the locus of smoothness of a scheme f*=0, where f* is a Weierstrass cubic formf*=f*(x,y,z)=y²z+a₁xyz+a₂yz²-x³-a₂x²z-a₄xz²-a₆z³, in which the coefficient a_i is a homogeneous polynomial with integral coefficients, of degree i in the ten coefficients of f, which we give explicitly. A key ingredint of the proof is a characterization, over sufficiently nice bases, of group algebraic spaces which can be described by such a Weierstrass equation.     

Adaptive regularization with cubics on manifolds

Mathematical Programming - Adaptive regularization with cubics (ARC) is an algorithm for unconstrained, non-convex optimization. Akin to the trust-region method, its iterations can be thought of as...     

On quartics with three-divisible sets of cusps

 We study the geometry and codes of quartic surfaces with many cusps. We apply Gr?bner bases to find examples of various configurations of cusps on quartics. Received: 10 May 2002 / Revised version: 11 November 2002 Published online: 3 March 2003 Permanent address: Institute of Mathematics, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, Poland. e-mail: rams@mi.uni-erlangen.de, rams@im.uj.edu.pl Research partially supported by the Schwerpunktprogramm ``Global methods in complex geometry' of the Deutsche Forschungsgemeinschaft, and by EAGER. Mathematics Subject Classification (2000): 14J25, 14J17     

Entirely circular quartics in the pseudo-Euclidean plane

A curve in the pseudo-Euclidean plane is circular if it passes through at least one of the absolute points. If it does not share any point with the absolute line except the absolute points, it is said to be entirely circular.     

Topological classification of real three-dimensional cubics

Mathematical Notes - The paper is devoted to finding topological types of nonsingular real three-dimensional cubics. It is proved that the following topological types exist: the projective space,...