Related papers: Computing Galois groups of Fano problems

Let K/F be a cyclic field extension of odd prime degree. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F) such that corresponding normal subgroups are indecomposable Fp[Gal(K/F)]-modules. For these…

We show that Fano lattice polygons define a class of balanced quivers with interesting properties. The combinatorics of these quivers is related to singularities of the underlying toric Fano surface. This allows us to show that every Fano…

We first review some topics in the classical computational geometry of lines, in particular the O(n^{3+\epsilon}) bounds for the combinatorial complexity of the set of lines in R^3 interacting with $n$ objects of fixed description…

We study (Galois) linear complementary dual codes over mixed alphabets arising from finite chain rings. We give a characterization of when a given code is of We study (Galois) linear complementary dual codes over mixed alphabets arising…

In this paper we consider the Galois covers of algebraic surfaces of degree 6, with all associated planar degenerations. We compute the fundamental groups of those Galois covers, using their degeneration. We show that for 8 types of…

We consider in detail an approach (proposed by the author earlier) where quantum states are described by elements of a linear space over a Galois field, and operators of physical quantities - by linear operators in this space. The notion of…

In this paper, we study the explicit geometry of threefolds, in particular, Fano varieties. We find an explicitly computable positive integer $N$, such that all but a bounded family of Fano threefolds have $N$-complements. This result has…

We show all the possible structures of finite subgroups of the automorphism groups of elliptic curves. Here the automorphism means the biholomorphic transformation. Using the result, we determine every Galois group G at outer Galois point…

This is a blend of two informal reports on the activities of the seminar on Galois representations and mirror symmetry given at the Conference on classification problems and mirror duality at the Steklov Institute, in March 2006, and at the…

We work over an algebraically closed field of positive characteristic. This paper investigates linear representations of Galois groups arising from wild Galois points on projective hypersurfaces. We prove that these Galois groups lift to…

If $\mathfrak{g} \subseteq \mathfrak{h}$ is an extension of Lie algebras over a field $k$ such that ${\rm dim}_k (\mathfrak{g}) = n$ and ${\rm dim}_k (\mathfrak{h}) = n + m$, then the Galois group ${\rm Gal} \, (\mathfrak{h}/\mathfrak{g})$…

We study Galois actions on $2+1$D topological quantum field theories (TQFTs), characterizing their interplay with theory factorization, gauging, the structure of gapped boundaries and dualities, 0-form symmetries, 1-form symmetries, and…

This paper initiates the systematic study of the number of points of bounded height on symmetric squares of weak Fano varieties. We provide a general framework for establishing the point count on $\text{Sym}^2 X$. In the specific case of…

In this preprint we present an outline of the multidimensional version of topological Galois theory. The theory studies topological obstruction to solvability of equations "in finite terms" (i.e. to their solvability by radicals, by…

Let $f(x)=x^{12}+ax^6+b \in \mathbb{Q}[x]$ be an irreducible polynomial, $g_4(x)=x^4+ax^2+b$, $g_6(x)=x^6+ax^3+b$, and let $G_4$ and $G_6$ be the Galois group of $g_4(x)$ and $g_6(x)$, respectively. Building upon known characterizations of…

The Galois group of the minimal polymonal of a Jones index value gives a new type of obstruction to a principal graph, thanks to a recent result of P.Etingof, D.Nikshych, and V.Ostrik. We show that the sequence of the graphs given by…

We give an explicit description of the arithmetic-geometric extension of iterated Galois groups of rational functions. This yields a complete solution to the extension problem when either the arithmetic or the geometric iterated Galois…

We develop algorithms to compute the differential Galois group corresponding to a one-parameter family of second order homogeneous ordinary linear differential equations with rational function coefficients. More precisely, we consider…

We study the deformation theory of the Fano scheme $\mathrm{F}=\mathrm{F}(\mathrm{X})$ of lines on a cubic $\mathrm{X}$ of dimension $d$ with only finitely many singularities. By taking the relative Fano scheme, we define a morphism…

We determine the absolute differential Galois group of the field $\mathbb{C}(x)$ of rational functions: It is the free proalgebraic group on a set of cardinality $|\mathbb{C}|$. This solves a longstanding open problem posed by B.H. Matzat.…