Related papers: Conic modules, secondary fans and non-commutative …

We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem…

We use convex polyhedral cones to study a large class of multivariate meromorphic germs, namely those with linear poles, which naturally arise in various contexts in mathematics and physics. We express such a germ as a sum of a holomorphic…

We study criteria for a ring - or more generally, for a small category - to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to…

Let $V$ be a plane smooth cubic curve over a finitely generated field $k.$ The Mordell-Weil theorem for $V$ states that there is a finite subset $P\subset V(k)$ such that the whole $V(k)$ can be obtained from $P$ by drawing secants and…

We construct and study the moduli of hypersurfaces in toric orbifolds. Let $X$ be a projective toric orbifold and $\alpha \in Cl(X)$ an ample class. The moduli space is constructed as a quotient of the linear system $|\alpha|$ by $G =…

The cluster-tilted algebras have been introduced by Buan, Marsh and Reiten, they are the endomorphism rings of cluster-tilting objects $T$ in cluster categories; we call such an algebra cluster-concealed in case $T$ is obtained from a…

We study rational modules over complete path and monomial algebras, and the problem of when rational modules over the dual $C^*$ of a coalgebra $C$ are closed under extensions, equivalently, when is the functor $Rat$ a torsion functor. We…

Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this thesis we explore this correspondence to classify smooth lattice…

We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients…

In the case of toric varieties, we continue the pursuit of Kontsevich's fundamental insight, Homological Mirror Symmetry, by unifying it with the Mori program. We give a refined conjectural version of Homological Mirror Symmetry relating…

We show the problem of counting homomorphisms from the fundamental group of a homology $3$-sphere $M$ to a finite, non-abelian simple group $G$ is #P-complete, in the case that $G$ is fixed and $M$ is the computational input. Similarly,…

We establish a link between rational homotopy theory and the problem which vector bundles admit complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively…

Conditions on the Koszul complex of a noetherian local ring $R$ guarantee that $\mathrm{Tor}^{R}_{i}(M,N)$ is non-zero for infinitely many $i$, when $M$ and $N$ are finitely generated $R$-modules of infinite projective dimension. These…

We prove existence of non-commutative crepant resolutions (in the sense of van den Bergh) of quotient singularities by finite and linearly reductive group schemes in positive characteristic. In dimension two, we relate these to resolutions…

We show that if a class of modules is closed under pure quotients, then it is precovering if and only if it is covering, and this happens if and only if it is closed under direct sums. This is inspired by a dual result by Rada and…

Let G be a finite abelian group. This paper is concerned with nonnegative functions on G that are sparse with respect to the Fourier basis. We establish combinatorial conditions on subsets S and T of Fourier basis elements under which…

Motivated from the theory of Hilbert-Schmidt morphisms between Hilbert C*-modules over commutative C*-algebras by Stern and van Suijlekom \textit{[J. Funct. Anal., 2021]}, we introduce the notion of p-absolutely summing morphisms between…

We study the topology of toric maps. We show that if $f\colon X\to Y$ is a proper toric morphism, with $X$ simplicial, then the cohomology of every fiber of $f$ is pure and of Hodge-Tate type. When the map is a fibration, we give an…

For Gorenstein quotient spaces $C^d/G$, a direct generalization of the classical McKay correspondence in dimensions $d\geq 4$ would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not…

Noncommutative (NC) sphere is introduced as a quotient of the enveloping algebra of the Lie algebra su(2). Using the Cayley-Hamilton identities we introduce projective modules which are analogues of line bundles on the usual sphere (we call…