Related papers: A Question about Pic(X) as a G-module

Let $X$ be an abstract tropical curve and let $G$ be a finite subgroup of the automorphism group of $X$. Let $D$ be a divisor on $X$ whose equivalence class is $G$-invariant. We address the following question: is there always a divisor $D'$…

Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parametrizing isomorphism classes of geometric objects (vector bundles, polarized…

Let $X$ be a variety with an action by an algebraic group $G$. In this paper we discuss various properties of $G$-equivariant $D$-modules on $X$, such as the decompositions of their global sections as representations of $G$ (when $G$ is…

In this note we address the following kind of question: let X be a smooth, irreducible, projective surface and D a divisor on X$satisfying some sort of positivity hypothesis, then is there some multiple of D depending only on X which is…

The ring of invariant polynomials ${\mathbb C}[V]^G$ over a given finite dimensional representation space $V$ of a complex reductive group $G$ is known, by a famous theorem of Hilbert, to be finitely generated. The general proof being…

Let $f:X \to S$ be a Galois cover of Riemann surfaces, with Galois group $G$. In this paper we analyze the $G$-invariant divisors on $X$, and their associated spaces of meromorphic functions, differentials, and $q$-differentials. We…

Let k be a perfect field and A a finite dimensional k-algebra of finite global dimension (e.g. the path algebra of a finite quiver without oriented cycles). Making use of the recent theory of noncommutative motives, we prove that the value…

Let X be a complex affine curve (not isomorphic to the affine line), and let Pic(D) be the group of autoequivalences of the category of D(X)-modules. Cannings and Holland have shown that Pic(D) fits into an exact sequence in which the other…

For any finite group G with a finite G-set X and a modular tensor category C we construct a part of the algebraic structure of an associated G-equivariant monoidal category: For any group element g in G we exhibit the module category…

Suppose $X$ is a smooth projective geometrically irreducible curve over a perfect field $k$ of positive characteristic $p$. Let $G$ be a finite group acting faithfully on $X$ over $k$ such that $G$ has non-trivial, cyclic Sylow…

Let $p$ be a prime number, $G$ be a finite $p$-group and $K$ be a field of characteristic $p$. The Modular Isomorphism Problem (MIP) asks whether the group algebra $KG$ determines the group $G$. Dealing with MIP, we investigated a question…

Let $F$ be a differential field of characteristic zero with algebraically closed constant field $C$. Let $E$ be a Picard--Vessiot closure of $F$, $R \subset E$ its Picard--Vessiot ring and $\Pi$ the differential Galois group of $E$ over…

The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in the moduli space of…

For a hyperelliptic curve of genus $g$, a divisor in general position of degree $g+1$ is given by polynomial equations. There is an action from an algebraic group on the representations of divisors by polynomials which fixes divisor…

We investigate infinite dimensional modules for an affine group scheme $\mathbb G$ of finite type over a field of positive characteristic $p$. For any subspace $X \subset \mathcal O(\mathbb G)$ of the coordinate algebra of $\mathbb G$, we…

If V is a representation of a linear algebraic group G, a set S of G-invariant regular functions on V is called separating if the following holds: If two elements v,v' from V can be separated by an invariant function, then there is an f…

Every action of a finite group scheme $G$ on a variety admits a projective equivariant model, but not necessarily a normal one. As a remedy, we introduce and explore the notion of $G$-normalization. In particular, every curve equipped with…

In this paper for each $n\ge g\ge 0$ we consider the moduli stack $\widetilde{\mathcal U}^{ns}_{g,n}$ of curves $(C,p_1,\ldots,p_n,v_1,\ldots,v_n)$ of arithmetic genus $g$ with $n$ smooth marked points $p_i$ and nonzero tangent vectors…

We give an answer to the abstract Capelli problem: Let $(G, V)$ be a multiplicity-free finite-dimensional representation of a connected reductive complex Lie group $G$ and $G'$ be its derived subgroup. Assume that the categorical quotient…

We study isolated points on the modular curves $X_{H}$, for $H$ a subgroup of $\operatorname{GL}_{2}(\mathbb{Z}/n \mathbb{Z})$ for some $n \geq 1$. In particular, we prove a single-sink theorem for such isolated points, which traces the…