Academic literature on the topic 'Moduli space'

A conjecture of H. Hopf states that if x(M²n) is a closed, Riemannian manifold of nonpositive sectional curvature, then its Euler characteristic x(M²n), should satify (-1)n x(M²n)≥ 0. Ruth Charney and Michael Davis investigated the conjecture in the context of piecewise Euclidean manifolds having "nonpositive curvature" in the sense of Gromov's CAT(0) inequality. In that context the conjecture can be reduced to a local version which predicts the sign of a "local Euler characteristic" at each vertex. They stated precisely various conjectures in their paper which we are interested in one of them stated as Conjecture D (see [1]) which is equivalent to the Hopf Conjecture for piecewise Euclidean manifolds cellulated by cubes. The goal of this thesis is to study the Charney - Davis Conjecture stated as Conjecture (D) by using sheaves on fans.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.<br>Includes bibliographical references (p. 57-59).<br>We define a moduli functor parametrizing finite maps from a projective (locally) Cohen-Macaulay curve to a fixed projective space. The definition of the functor includes a number of technical conditions, but the most important is that the map is almost everywhere an isomorphism onto its image. The motivation for this definition comes from trying to interpolate between the Hilbert scheme and the Kontsevich mapping space. The main result of this thesis is that our functor is represented by a proper algebraic space. As an application we obtain interesting compactifications of the spaces of smooth curves in projective space. We illustrate this in the case of rational quartics, where the resulting space appears easier than the Hilbert scheme.<br>by Morten Oskar Hønsen.<br>Ph.D.

Oort gave a complete description of symplectic commutative group schemes killed by p and of rank p2g . Each such group appears as the p-torsion group scheme of some principally polarized abelian variety and this classification can be given in terms of final sequences. In this thesis, we focus on the particular situation where the abelian variety is the Jacobian of a hyperelliptic curve. We concentrate on describing the subspace of the moduli space of hyperelliptic curves, or rather the cycle, corresponding to a given final sequence. Especially, we concentrate on describing the subspace corresponding to the non-ordinary locus, which is a union of final sequences.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (p. 43-46).<br>The moduli space of curves has proven itself a central object in algebraic geometry. The past decade has seen substantial progress in understanding its geometry. This has been spurred by a flurry of ideas from geometry (algebraic, symplectic, and differential), topology, combinatorics, and physics. One way of understanding its birational geometry is by describing its cones of ample and effective divisors and the dual notion of the Mori cone (the closed cone of curves). This thesis aims at giving a brief introduction to the moduli space of n-pointed stable curves of genus ... and some intuition into it and its structure. We do so by surveying what is currently known about the ample and the effective cones of ... , and the problem of determining the closed cone of curves ... The emphasis in this exposition lies on a partial resolution of the Fulton-Faber conjecture (the F-conjecture). Recently, some positive results were announced and the conjecture was shown to be true in a select few cases. Conjecturally, the ample cone has a very simple description as the dual cone spanned by the F-curves. Faber curves (or F-curves) are irreducible components of the locus in ... that parameterize curves with 3g - 4 + n nodes. There are only finitely many classes of F-curves. The conjecture has been verified for the moduli space of curves of small genus. The conjecture predicts that for large g, despite being of general type, ... behaves from the point of view of Mori theory just like a Fano variety. Specifically, this means that the Mori cone of curves is polyhedral, and generated by rational curves. It would be pleasantly surprising if the conjecture holds true for all cases. In the case of the effective cone of divisors the situation is more complicated. F-conjecture. A divisor on ... is ample (nef) if and only if it intersects positively (nonnegatively) all 1-dimensional strata or the F-curves . In other words, every extremal ray of the Mori cone of effective curves NE1(Mg,n) is generated by a one dimensional stratum. The main results presented here are: (i) the Mori cone ... is generated by F-curves when ...<br>by Shashank S. Dwivedi.<br>S.M.