625 research outputs found

    Z-Pencils

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    The matrix pencil (A,B) = {tB-A | t \in C} is considered under the assumptions that A is entrywise nonnegative and B-A is a nonsingular M-matrix. As t varies in [0,1], the Z-matrices tB-A are partitioned into the sets L_s introduced by Fiedler and Markham. As no combinatorial structure of B is assumed here, this partition generalizes some of their work where B=I. Based on the union of the directed graphs of A and B, the combinatorial structure of nonnegative eigenvectors associated with the largest eigenvalue of (A,B) in [0,1) is considered.Comment: 8 pages, LaTe

    Linear pencils encoded in the Newton polygon

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    Let CC be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon Δ\Delta. It is classical that the geometric genus of CC equals the number of lattice points in the interior of Δ\Delta. In this paper we give similar combinatorial interpretations for the gonality, the Clifford index and the Clifford dimension, by removing a technical assumption from a recent result of Kawaguchi. More generally, the method shows that apart from certain well-understood exceptions, every base-point free pencil whose degree equals or slightly exceeds the gonality is 'combinatorial', in the sense that it corresponds to projecting CC along a lattice direction. We then give an interpretation for the scrollar invariants associated to a combinatorial pencil, and show how one can tell whether the pencil is complete or not. Among the applications, we find that every smooth projective curve admits at most one Weierstrass semi-group of embedding dimension 22, and that if a non-hyperelliptic smooth projective curve CC of genus g≥2g \geq 2 can be embedded in the nnth Hirzebruch surface Hn\mathcal{H}_n, then nn is actually an invariant of CC.Comment: This covers and extends sections 1 to 3.4 of our previously posted article "On the intrinsicness of the Newton polygon" (arXiv:1304.4997), which will eventually become obsolete. arXiv admin note: text overlap with arXiv:1304.499

    New moduli spaces of pointed curves and pencils of flat connections

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    It is well known that formal solutions to the Associativity Equations are the same as cyclic algebras over the homology operad (H∗(Mˉ0,n+1))(H_*(\bar{M}_{0,n+1})) of the moduli spaces of nn--pointed stable curves of genus zero. In this paper we establish a similar relationship between the pencils of formal flat connections (or solutions to the Commutativity Equations) and homology of a new series Lˉn\bar{L}_n of pointed stable curves of genus zero. Whereas Mˉ0,n+1\bar{M}_{0,n+1} parametrizes trees of P1\bold{P}^1's with pairwise distinct nonsingular marked points, Lˉn\bar{L}_n parametrizes strings of P1\bold{P}^1's stabilized by marked points of two types. The union of all Lˉn\bar{L}_n's forms a semigroup rather than operad, and the role of operadic algebras is taken over by the representations of the appropriately twisted homology algebra of this union.Comment: 37 pages, AMSTex. Several typos corrected, a reference added, subsection 3.2.2 revised, subsection 3.2.4 adde

    Extended modular operad

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    This paper is a sequel to [LoMa] where moduli spaces of painted stable curves were introduced and studied. We define the extended modular operad of genus zero, algebras over this operad, and study the formal differential geometric structures related to these algebras: pencils of flat connections and Frobenius manifolds without metric. We focus here on the combinatorial aspects of the picture. Algebraic geometric aspects are treated in [Ma2].Comment: 38 pp., amstex file, no figures. This version contains additional references and minor change
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