Related papers: Modules over the Noncommutative Torus and Elliptic…

We show how quiver representations and their invariant theory natu- rally arise in the study of some moduli spaces parametrizing bundles dened on an algebraic curve, and how they lead to ne results regarding the geometry of these spaces.

It is well-known that del Pezzo surfaces of degree $9-n$ one-to-one correspond to flat $E_n$ bundles over an elliptic curve. In this paper, we construct $ADE$ bundles over a broader class of rational surfaces which we call $ADE$ surfaces,…

The Hodge bundle $\omega$ over a modular curve is a square-root of the canonical bundle twisted by the cuspidal divisor, or a theta characteristic, due to the Kodaira--Spencer isomorphism. We prove that, in most cases, a section of a theta…

In the spirit of the geometric approach to two-dimensional conformal field theory, we explicitly associate to every holomorphic vertex operator algebra a section of a power of Hodge line bundle on the moduli space of curves of arbitrary…

For any non-simply laced Lie group $G$ and elliptic curve $\Sigma$, we show that the moduli space of flat $G$ bundles over $\Sigma$ can be identified with the moduli space of rational surfaces with $G$-configurations which contain $\Sigma$…

We exhibit a smooth compactification of the moduli space of elliptic curves in a product of projective spaces with tangency along a subset of its toric boundary divisors. This is a Vakil--Zinger type of desingularization for maps to a…

Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, it admits a surjection from a modular curve $X_0(N) \to \mathsf{E}$, and the minimal degree among such maps is called the modular degree of $\mathsf{E}$. By the…

The class of support $\tau$-tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article we study $\tau$-tilting finite algebras, i.e. finite dimensional…

We address quantization of the natural symplectic structure on a moduli space of parabolic vector bundles of parabolic degree zero over $\mathbf{CP}^1$ with four parabolic points and parabolic weights in {0,1/2}. Identifying such parabolic…

Let $X$ be any smooth simply connected projective surface. We consider some moduli space of pure sheaves of dimension one on $X$, i.e. $\mhu$ with $u=(0,L,\chi(u)=0)$ and $L$ an effective line bundle on $X$, together with a series of…

We construct vector-valued modular forms on moduli spaces of curves and abelian varieties using effective divisors in projectivized Hodge bundles over moduli of curves. Cycle relations tell us the weight of these modular forms. In…

In this paper we study noncommutative plane curves, i.e. non-commutative k-algebras for which the 1-dimensional simple modules form a plane curve. We study extensions of simple modules and we try to enlighten the completion problem, i.e.…

In this paper we give a survey about the classification of vector bundles and torsion free sheaves on degenerations of elliptic curves. Coherent sheaves on singular curves of arithmetic genus one can be studied using the technique of matrix…

This is a copy of the article published in IMRN (2007). I describe the noncommutative Batalin-Vilkovisky geometry associated naturally with arbitrary modular operad. The classical limit of this geometry is the noncommutative symplectic…

We analyze in detail projective modules over two-dimensional noncommutative tori and complex structures on these modules.We concentrate our attention on properties of holomorphic vectors in these modules; the theory of these vectors…

We prove finite generation of the algebra of type A conformal blocks over arbitrary stable curves of any genus. As an application we construct a flat family of irreducible normal projective varieties over the moduli stack of stable pointed…

The tau function on the moduli space of generic holomorphic 1-differentials on complex algebraic curves is interpreted as a section of a line bundle on the projectivized Hodge bundle over the moduli space of stable curves. The asymptotics…

The gauge formulation of Einstein gravity in AdS$_3$ background leads to a boundary theory that breaks modular symmetry and loses the covariant form. We examine the Weyl anomaly for the cylinder and torus manifolds. The divergent term is…

We show that any commutative rationally ruled surface with a choice of anticanonical curve admits a 1-parameter family of noncommutative deformations parametrized by the Jacobian of the anticanonical curve, and show that many standard facts…

We determine explicit birational models over Q for the modular surfaces parametrising pairs of N-congruent elliptic curves in all cases where this surface is an elliptic surface. In each case we also determine the rank of the Mordell-Weil…