Rank 2 vector bundles on P4 with c1 odd and contact curves.
Rank 4 vector bundles on the quintic threefold
By the results of the author and Chiantini in [3], on a general quintic threefold X⊂P 4 the minimum integer p for which there exists a positive dimensional family of irreducible rank p vector bundles on X without intermediate cohomology is at least three. In this paper we show that p≤4, by constructing series of positive dimensional families of rank 4 vector bundles on X without intermediate cohomology. The general member of such family is an indecomposable bundle from the extension class Ext 1...
Rank additivity for quasi-tilted algebras of canonical type
Given the category of coherent sheaves over a weighted projective line (of any representation type), the endomorphism ring of an arbitrary tilting sheaf - which is by definition an almost concealed canonical algebra - is shown to satisfy a rank additivity property (Theorem 3.2). Moreover, this property extends to the representationinfinite quasi-tilted algebras of canonical type (Theorem 4.2). Finally, it is demonstrated that rank additivity does not generalize to the case of tilting complexes...
Rank frequencies for quadratic twists of elliptic curves.
Rank of elliptic curves associated to Brahmagupta quadrilaterals
We construct a family of elliptic curves with six parameters, arising from a system of Diophantine equations, whose rank is at least five. To do so, we use the Brahmagupta formula for the area of cyclic quadrilaterals (p³,q³,r³,s³) not necessarily representing genuine geometric objects. It turns out that, as parameters of the curves, the integers p,q,r,s along with the extra integers u,v satisfy u⁶+v⁶+p⁶+q⁶ = 2(r⁶+s⁶), uv = pq, which, by previous work, has infinitely many integer solutions.
Rank two Ulrich modules over the affine cone of the simple node.
Rank two vector bundles over regular elliptic surfaces.
Rank zero quadratic twists of modular elliptic curves
Rank-2 Vector Bundles on a Ruled Surface. I.
Rank-2 Vector Bundles on a Ruled Surface II.
Rank-3 stable bundles on rational rundle ruled surfaces.
Ranks of quadratic twists of elliptic curves
We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability to find...
Rank-two vector bundles on general quartic hypersurfaces in P4.
In this paper all non-splitting rank-two vector bundles E without intermediate cohomology on a general quartic hypersurface X in P4 are classified. In particular, the existence of some curves on a general quartic hypersurface is proved.
Rank-two vector bundles on Hirzebruch surfaces
We survey some parts of the vast literature on vector bundles on Hirzebruch surfaces, focusing on the rank-two case.
Rational 1- and 2-cuspidal plane curves.
Rational and Generic Cohomology
Rational and homological equivalence for real cycles.
Rational approximation to real points on conics
A point with coordinates in a subfield of of transcendence degree one over , with linearly independent over , may have a uniform exponent of approximation by elements of that is strictly larger than the lower bound given by Dirichlet’s box principle. This appeared as a surprise, in connection to work of Davenport and Schmidt, for points of the parabola . The goal of this paper is to show that this phenomenon extends to all real conics defined over , and that the largest exponent of...
Rational Bézier curves with infinitely many integral points
In this paper we consider rational Bézier curves with control points having rational coordinates and rational weights, and we give necessary and sufficient conditions for such a curve to have infinitely many points with integer coefficients. Furthermore, we give algorithms for the construction of these curves and the computation of theirs points with integer coefficients.
Rational components of Hilbert schemes