Related papers: Projective Self-dual polygons in higher dimensions

In this article we address the problem of computing the dimension of the space of plane curves of degree $d$ with $n$ general points of multiplicity $m$. A conjecture of Harbourne and Hirschowitz implies that when $d \geq 3m$, the dimension…

Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a…

We study the local geometry of the moduli space of intermediate Jacobians of $(2,2)$-threefolds in ${\mathbb P}^2 \times {\mathbb P}^2$. More precisely, we prove that a composition of the second fundamental form of the Siegel metric in…

A conjectural recursive relation for the Poincar\'e polynomial of the Hitchin moduli space is derived from wallcrossing in the refined local Donaldson-Thomas theory of a curve. A doubly refined generalization of this theory is also…

Let $V$ be a two-dimensional vector space over a field $\mathbb F$ of characteristic not $2$ or $3$. We show there is a canonical surjection $\nu$ from the set of suitably generic commutative algebra structures on $V$ modulo the action of…

Let $\Gamma$ be a dual polar graph with diameter $D \geqslant 3$, having as vertices the maximal isotropic subspaces of a finite-dimensional vector space over the finite field $\mathbb{F}_q$ equipped with a non-degenerate form (alternating,…

We present several results on the geometry of the quantum projective plane CP2q. They include: explicit generators for the K-theory and the K-homology; a real calculus with a Hodge star operator; anti-selfdual connections on line bundles…

P. Clarke describes mirror symmetry as a duality between Landau--Ginzburg models, so that the dual of an LG model is another LG model. We describe examples in which the underlying space is a total space of a vector bundle on the projective…

Given two tensor fields of type (1,1) on a smooth n-dimensional manifold M, such that all their Fr\"olicher-Nijenhuis brackets vanish, the algebra of differential forms on M becomes a bi-differential graded algebra. As a consequence, there…

In this work, we study Lagrangian formulations for self-dual gauge theories, also known as chiral $n$-form gauge theories, for $n = 2p$ in $D = 4p+2$ dimensional spacetime. Motivated by a recent formulation of M5-branes derived from the BLG…

We show that totally geodesic subvarieties of the moduli space $\mathcal M_{g,n}$ of genus $g$ curves with $n$ marked points, endowed with the Weil--Petersson metric, are locally rigid. This implies that covering constructions -- examples…

Given a combinatorial triangulation of an $n$-gon, we study (a) the space of all possible drawings in the plane such the edges are straight line segments and the boundary has a fixed shape, and (b) the algebraic variety of possibilities for…

We propose definitions of complex manifolds $\mathcal{P}_M(X,m,n)$ that could potentially be used to construct the symplectic Khovanov homology of $n$-stranded links in lens spaces. The manifolds $\mathcal{P}_M(X,m,n)$ are defined as moduli…

Regular maps on linear fractional groups $PSL(2,q)$ and $PGL(2,q$) have been studied for many years and the theory is well-developed, including generating sets for the asscoiated groups. This paper studies the properties of self-duality,…

Let X be a geometrically irreducible smooth projective curve over a field k. We describe the algebra of endomorphisms of indecomposable unstable vector bundles over X of rank 2 and degree d. Fixing some numerical invariants, namely the…

We describe the K-moduli spaces of weighted hypersurfaces of degree $2(n+3)$ in $\mathbb{P}(1,2,n+2,n+3)$. We show that the K-polystable limits of these weighted hypersurfaces are also weighted hypersurfaces of the same degree in the same…

We compute the automorphism group of the dual complex $\mathsf{T}_{d, n}$ of the boundary divisor in the Kontsevich moduli space $\overline{\mathcal{M}}_{0, n}(\mathbb{P}^r, d)$. When $d \geq 2$, we find that $\mathrm{Aut}(\mathsf{T}_{d,…

We explicitly describe a structure of a regular cell complex $K(L)$ on the moduli space $M(L)$ of a planar polygonal linkage $L$. The combinatorics is very much related (but not equal) to the combinatorics of the permutahedron. In…

Given an orthogonal bundle $E$ over a smooth projective curve $X$ we define a Hecke transformation in the moduli space of orthogonal bundles by performing an elementary transformation with respect to a Lagrangian submodule $L \subset…

We study Le Potier's strange duality conjecture on $\mathbb{P}^2$. We focus on the strange duality map $SD_{c_n^r,d}$ which involves the moduli space of rank $r$ sheaves with trivial first Chern class and second Chern class $n$, and the…