arXiv subjects

Marton Hablicsek

Publications and source records attributed to Marton Hablicsek.

8 recordsLinked to original sources

arXiv2026

Logarithmic Hochschild (co)homology of logarithmic orbifolds

Recently, the authors of this paper introduced logarithmic Hochschild (co)homology of logarithmic spaces in a geometric way using formality of derived intersections. In this paper, the authors extend the decomposition theorem for the logarithmic Hochschild (co)homology of firm orbifolds to general logarithmic orbifolds and consider two applications of the decomposition theorem. First, we consider two versions of a symmetric product and compute the logarithmic Hochschild homology of them. Second, we show that logarithmic Hochschild homology is invariant under root stack operations.

math.AG↗

arXiv2021

Weak $E_2$-Morita equivalences via quantization of the 1-shifted cotangent bundle

In this paper, we investigate the structure of the convergent quantization of the 1-shifted cotangent bundle $S$ of a smooth scheme $X$ over a perfect field of positive characteristic. The quantization is an $E_2$-algebra over the Frobenius twist $S'$ of the 1-shifted cotangent bundle which restricted to the zero section $X'\rightarrow S'$ is weakly $E_2$-Morita equivalent to the structure sheaf $\mathscr O_{X'}$ of the Frobenius twist $X'$ of $X$. Explicitly, we show that the $(\infty,2)$-category of coherent (left-)modules over $\mathscr O_{X'}$ is equivalent to the full subcategory of the $(\infty,2)$-category of coherent (left-)modules over the quantization restricted to the zero section generated by $\mathscr O_X$ are equivalent.

math.AG↗

arXiv2021

On the number of rich lines in high dimensional real vector spaces

In this short note we use the polynomial partitioning lemma to strengthen a recent result of Dvir and Gopi about the number of rich lines in high dimensional Euclidean spaces. Our result shows that if there are sufficiently many rich lines incident to a set of points then large fraction of them must be contained in a hyperplane.

math.CO↗

arXiv2020

Algebraic 3D Graphic Statics: Constrained Areas

This research provides algorithms and numerical methods to geometrically control the magnitude of the internal and external forces in the reciprocal diagrams of 3D/Polyhedral Graphic statics (3DGS). In 3DGS, the form of the structure and its equilibrium of forces is represented by two polyhedral diagrams that are geometrically and topologically related. The areas of the faces of the force diagram represent the magnitude of the internal and external forces in the system. For the first time, the methods of this research allow the user to control and constrain the areas and edge lengths of the faces of general polyhedrons that can be convex, self-intersecting, or concave. As a result, a designer can explicitly control the force magnitudes in the force diagram and explore the equilibrium of a variety of compression and tension-combined funicular structural forms. In this method, a quadratic formulation is used to compute the area of a single face based on its edge lengths. The approach is applied to manipulating the face geometry with a predefined area and the edge lengths. Subsequently, the geometry of the polyhedron is updated with newly changed faces. This approach is a multi-step algorithm where each step includes computing the geometry of a single face and updating the polyhedral geometry. One of the unique results of this framework is the construction of the zero-area, self-intersecting faces, where the sum of the signed areas of a self-intersecting face is zero, representing a member with zero force in the form diagram. The methodology of this research can clarify the equilibrium of some systems that could not be previously justified using reciprocal polyhedral diagrams. Therefore, it generalizes the principle of the equilibrium of polyhedral frames and opens a completely new horizon in the design of highly-sophisticated funicular polyhedral structures beyond compression-only systems.

cs.CG↗

arXiv2020

Algebraic geometry over the residue field of the infinite place

Nikolai Durov introduced the theory of generalized rings and schemes to study Arakelov geometry in an alternative algebraic framework, and introduced the residue field at the infinite place. We show an elementary algebraic approach to modules and algebras over this object, define prime congruences, show that the polynomial ring of n variables is of Krull dimension n, and derive a prime decomposition theorem for these primes.

math.RA↗

arXiv2016

Derived intersections and the Hodge theorem

The algebraic Hodge theorem was proved in a beautiful 1987 paper by Deligne and Illusie, using positive characteristic methods. We argue that the central algebraic object of their proof can be understood geometrically as a line bundle on a derived scheme. In this interpretation, the Deligne-Illusie result can be seen as a proof that this line bundle is trivial under certain assumptions. We give a criterion for the triviality of this line bundle in a more general context. The proof uses techniques from derived algebraic geometry, specifically arguments which show the formality of certain derived intersections. Applying our criterion we recover Deligne and Illusie's original result. We also apply these techniques to the result of Barannikov-Kontsevich, Sabbah, and Ogus-Vologodsky concerning the formality of the twisted de Rham complex.

math.AG↗

arXiv2014

Formality of derived intersections and the orbifold HKR isomorphism

We study when the derived intersection of two smooth subvarieties of a smooth variety is formal. As a consequence we obtain a derived base change theorem for non-transversal intersections. We also obtain applications to the study of the derived fixed locus of a finite group action and argue that for a global quotient orbifold the exponential map is an isomorphism between the Lie algebra of the free loop space and the loop space itself. This allows us to give new proofs of the HKR decomposition of orbifold Hochschild (co)homology into twisted sectors.

math.AG↗

arXiv2013

An incidence conjecture of Bourgain over fields of positive characteristic

In this note we generalize a recent theorem of Guth and Katz on incidences between points and lines in $3$-space from characteristic $0$ to characteristic $p$, and we explain how some of the special features of algebraic geometry in characteristic $p$ manifest themselves in problems of incidence geometry.

math.CO↗