Stable vector bundles over cuspidal cubics

We give a complete classification of stable vector bundles over a cuspidal cubic and calculate their cohomologies. The technique of matrix problems is used, similar to [2, 3].     

The picard group of the variety of plane cuspidal cubics of ℙ3

We give a compactification of the varietyU of non-degenerate plane cuspidal cubics of ?³. We construct this compactification by means of the projective bundleX of a suitable vector bundleE. We describe the intersection ring ofX and, as a consequence, we obtain the intersection numbers ofU that satisfy 10 conditions of the following kinds:ρ, that the plane determined by the cuspidal cubic go through a point;c, that the cusp be on a plane;q, that the cuspidal tangent intersect a line;μ, that the cuspidal cubic intersect a line. Moreover, we prove that the Picard group of the varietyU is a product of two infinite cyclic groups generated byρ andc?q.     

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.     

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.     

On rational cuspidal curves

Let be a rational curve of degree d which has only one analytic branch at each point. Denote by m the maximal multiplicity of singularities of C. It is proven in [MS] that . We show that where is the square of the “golden section”. We also construct examples which show that this estimate is asymptotically sharp. When , we show that and this estimate is sharp. The main tool used here, is the logarithmic version of the Bogomolov-Miyaoka-Yau inequality. For curves as above we give an interpretation of this inequality in terms of the number of parameters describing curves of a given degree and the number of conditions imposed by singularity types. Received: 11 February 2000 / Published online: 8 November 2002 RID="*" ID="*" Partially supported by Grants RFFI-96-01-01218 and DGICYT SAB95-0502     

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)     

Geometrical properties of some Euler and circular cubics. Part 2

This paper is a continuation of Part 1, previously published under the same title. The reader is asked to refer back to Part 1 for the Glossary and references.     

Singular cubics of CP 2 as limits of sequences of tori

We use the modular invariant j to understand singular cubics of CP ² as limits of sequences of tori and we observe different behaviours according to the cubic type.      

Double cubics and double quartics

We study a double cover branched over a smooth divisor such that R is cut on V by a hypersurface of degree 2(n−deg(V)), where n ≥ 8 and V is a smooth hypersurface of degree 3 or 4. We prove that X is nonrational and birationally superrigid.     

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...     

The conservation relation for cuspidal representations

Let \({\pi \in Cusp ({\rm U}(V_n))}\) be a smooth cuspidal irreducible representation of a unitary group U(V _n ) of dimension n over a non-Archimedean locally compact field. Let \({W^\pm_m}\) be the two isomorphism classes of Hermitian spaces of dimension m, and the denote by \({\tau^+ \in Cusp ({\rm U}(W_{m^+}^+))}\) and \({\tau^- \in Cusp ({\rm U}(W_{m^-}^-))}\) the first non-zero theta lifts of π. In this article we prove that m ⁺ + m ^? = 2n + 2, which was conjectured in Harris et al. (J AMS 9:941–1004, 1996, Speculations 7.5 and 7.6). We prove similar equalities for the other dual pairs of type I: the symplectic-orthogonal dual pairs and the quaternionic dual pairs.     

Coset intersection of irreducible plane cubics

In a projective plane $PG(2,\mathbb K )$ over an algebraically closed field $\mathbb K $ of characteristic $p\ge 0$ , let $\Omega $ be a pointset of size $n$ with $5\le n \le 9$ . The coset intersection problem relative to $\Omega $ is to determine the family $\mathbf F$ of irreducible cubics in $PG(2,\mathbb K )$ for which $\Omega $ is a common coset of a subgroup of the additive group $(\mathcal F ,+)$ for every $\mathcal F \in \mathbf F$ . In this paper, a complete solution of this problem is given.     

Geometric Hermite interpolation with Tschirnhausen cubics

Explicit formulae are found that give the unique Tschirnhausen cubic that solves a geometric Hermite interpolation problem. That solution is used to create a planar G¹ spline by joining segments of Tschirnhausen cubics. If the geometric Hermite data is from a smooth function, the Tschirnhausen cubic approximates the smooth function. The error in the approximation of a short segment of length h can be expressed as a power series in h. The error is O(h⁴) and the coefficient of the leading term is found.