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We will discuss the moduli space of genus 0 stable maps to a low degree hypersurface X inside an Orthogonal Grassmannian OG(k,n). In particular we show that the space of lines in such a general hypersurface is irreducible of the expected dimension. Next, we study the space of chains of lines connecting two general points and obtain the result that such a space is rationally connected. Through a smoothing argument, we also obtain rationally connectedness for a component of the stable map space. Finally we construct a very twisting surface in a general such X and conclude rationally simply connectedness.
We study fibrations by curves on smooth projective threefolds. Our main result shows that given a smooth threefold , there exists an integer g0 = g0(X ) such that for every g ≥ g0, there is a birational model X admitting a morphism X′ → ℙ2 whose general fibre is a smooth curve of genus .
Abstract:
This dissertation constructs a new family of non-compact scalar-flat Kahler manifolds asymptotic to Calabi-Yau cones using a gluing method. The main result shows that if a manifold admits a scalar-flat Kahler metric, then its blow-up also admits a scalar-flat Kahler metric. The method follows from the work of Claudio Arezzo and Frank Pacard, who established analogous results in the compact setting. We extend their approach to the non-compact case. The key analytic ingredient is the bijectivity of the Lichnerowicz operator on weighted Holder spaces.
Speaker: Prabhat Devkota
Abstract:
We explore the geometry and topology of the moduli spaces of meromorphic differentials on curves of genus 0. In particular, we compute the cohomology with rational coefficients of the smooth orbifold compactification of these moduli spaces given by the multi-scale differentials. Additionally, we also determine all the cases where these moduli spaces are smooth varieties, and in these cases, we also compute their integral cohomology. In the case of special signature (0n , −2), we prove that the moduli space of multi-scale differentials is isomorphic to the classically studied space called the wonderful variety. Furthermore, in this special case, we relate it to a similar moduli space parameterizing meromorphic 1-forms on genus 0 curves, introduced by Halpern-Leistner and Robotis, called the moduli space of multiscale lines with collision.
