Plane curves with prescribed triple points: a toric approach

We will use toric degenerations of the projective plane ${{\mathbb{P}}^ 2}$ to give a new proof of the triple points interpolation problems in the projective plane. We also give a complete list of toric surfaces that are useful as components in this degeneration

Vanishing theorems for linearly obstructed divisors

We study divisors in the blow-up of $\mathbb{P}^n$ at points in general position that are non-special with respect to the notion of linear speciality introduced in [5]. We describe the cohomology groups of their strict transforms via the blow-up of the space along their linear base locus. We extend the result to non-effective divisors that sit in a small region outside the effective cone. As an application, we describe linear systems of divisors in $\mathbb{P}^n$ blown-up at points in star configuration and their strict transforms via the blow-up of the linear base locus

An invitation to 2D TQFT and quantization of Hitchin spectral curves

This article consists of two parts. In Part 1, we present a formulation of two-dimensional topological quantum field theories in terms of a functor from a category of Ribbon graphs to the endofuntor category of a monoidal category. The key point is that the category of ribbon graphs produces all Frobenius objects. Necessary backgrounds from Frobenius algebras, topological quantum field theories, and cohomological field theories are reviewed. A result on Frobenius algebra twisted topological recursion is included at the end of Part 1. In Part 2, we explain a geometric theory of quantum curves. The focus is placed on the process of quantization as a passage from families of Hitchin spectral curves to families of opers. To make the presentation simpler, we unfold the story using SL_2(\mathbb{C})-opers and rank 2 Higgs bundles defined on a compact Riemann surface $C$ of genus greater than $1$. In this case, quantum curves, opers, and projective structures in $C$ all become the same notion. Background materials on projective coordinate systems, Higgs bundles, opers, and non-Abelian Hodge correspondence are explained.Comment: 53 pages, 6 figure

Edge contraction on dual ribbon graphs and 2D TQFT

We present a new set of axioms for 2D TQFT formulated on the category of cell graphs with edge-contraction operations as morphisms. We construct a functor from this category to the endofunctor category consisting of Frobenius algebras. Edge-contraction operations correspond to natural transformations of endofunctors, which are compatible with the Frobenius algebra structure. Given a Frobenius algebra A, every cell graph determines an element of the symmetric tensor algebra defined over the dual space A*. We show that the edge-contraction axioms make this assignment depending only on the topological type of the cell graph, but not on the graph itself. Thus the functor generates the TQFT corresponding to A.Comment: accepted in Journal of Algebra (22 pages, 13 figures