10,308 research outputs found

    Common transversals and tangents to two lines and two quadrics in P^3

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    We solve the following geometric problem, which arises in several three-dimensional applications in computational geometry: For which arrangements of two lines and two spheres in R^3 are there infinitely many lines simultaneously transversal to the two lines and tangent to the two spheres? We also treat a generalization of this problem to projective quadrics: Replacing the spheres in R^3 by quadrics in projective space P^3, and fixing the lines and one general quadric, we give the following complete geometric description of the set of (second) quadrics for which the 2 lines and 2 quadrics have infinitely many transversals and tangents: In the nine-dimensional projective space P^9 of quadrics, this is a curve of degree 24 consisting of 12 plane conics, a remarkably reducible variety.Comment: 26 pages, 9 .eps figures, web page with more pictures and and archive of computations: http://www.math.umass.edu/~sottile/pages/2l2s

    Fundamental groups of some special quadric arrangements

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    In this work we obtain presentations of fundamental groups of the complements of three families of quadric arrangements in P2\mathbb{P}^2. The first arrangement is a union of nn quadrics, which are tangent to each other at two common points. The second arrangement is composed of nn quadrics which are tangent to each other at one common point. The third arrangement is composed of nn quadrics, n1n-1 of them are tangent to the nn'th one and each one of the n1n-1 quadrics is transversal to the other n2n-2 ones.Comment: 21 pages, 12 main figures, appears in Revista Mathematica

    Groebner bases for spaces of quadrics of codimension 3

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    Let R=i0RiR=\oplus_{i\geq 0} R_i be an Artinian standard graded KK-algebra defined by quadrics. Assume that dimR23\dim R_2\leq 3 and that KK is algebraically closed of characteristic 2\neq 2. We show that RR is defined by a Gr\"obner basis of quadrics with, essentially, one exception. The exception is given by K[x,y,z]/IK[x,y,z]/I where II is a complete intersection of 3 quadrics not containing the square of a linear form.Comment: Minor changes, to appear in the J. Pure Applied Algebr

    Billiard algebra, integrable line congruences, and double reflection nets

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    The billiard systems within quadrics, playing the role of discrete analogues of geodesics on ellipsoids, are incorporated into the theory of integrable quad-graphs. An initial observation is that the Six-pointed star theorem, as the operational consistency for the billiard algebra, is equivalent to an integrabilty condition of a line congruence. A new notion of the double-reflection nets as a subclass of dual Darboux nets associated with pencils of quadrics is introduced, basic properies and several examples are presented. Corresponding Yang-Baxter maps, associated with pencils of quadrics are defined and discussed.Comment: 18 pages, 8 figure

    Toric Ideals of Lattice Path Matroids and Polymatroids

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    We show that the toric ideal of a lattice path polymatroid is generated by quadrics corresponding to symmetric exchanges, and give a monomial order under which these quadrics form a Gr\"obner basis. We then obtain an analogous result for lattice path matroids.Comment: 9 pages, 4 figure

    Peterson's Deformations of Higher Dimensional Quadrics

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    We provide the first explicit examples of deformations of higher dimensional quadrics: a straightforward generalization of Peterson's explicit 1-dimensional family of deformations in C3\mathbb{C}^3 of 2-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere S2C3\mathbb{S}^2\subset\mathbb{C}^3 to an explicit (n1)(n-1)-dimensional family of deformations in C2n1\mathbb{C}^{2n-1} of nn-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere SnCn+1\mathbb{S}^n\subset\mathbb{C}^{n+1} and non-degenerate joined second fundamental forms. It is then proven that this family is maximal
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