We develop the representation theory of selfinjective algebras which admit Galois coverings by the repetitive algebras of algebras whose derived category of bounded complexes of finite-dimensional modules is equivalent to the derived category of coherent sheaves on a weighted projective line with virtual genus greater than one.

In the present paper, we give a first general construction of compactified moduli spaces for semistable $G$-bundles on an irreducible complex projective curve $X$ with exactly one node, where $G$ is a semisimple linear algebraic group over the complex numbers.

We consider the linear system $|2{\Theta }_0|$ of second order theta functions over the Jacobian $JC$ of a non-hyperelliptic curve $C$. A result by J.Fay says that a divisor $D\in |2{\Theta }_0|$ contains the origin $𝒪\in JC$ with multiplicity $4$ if and only if $D$ contains the surface $C-C=\left\{𝒪\right(p-q)\mid p,q\in C\}\subset JC$. In this paper we generalize Fay’s result and some previous work by R.C.Gunning. More precisely, we describe the relationship between divisors containing $𝒪$ with multiplicity $6$, divisors containing the fourfold $C_2-C_2=\{𝒪(p+q-r-s)\mid p,q,r,s\in C\}$, and divisors singular along $C-C$, using the third exterior...

The subject of this article is the notion of $r$-spin structure: a line bundle whose $r$th power is isomorphic to the canonical bundle. Over the moduli functor ${\mathsf{M}}_g$ of smooth genus-$g$ curves, $r$-spin structures form a finite torsor under the group of $r$-torsion line bundles. Over the moduli functor ${\overline{\mathsf{M}}}_g$ of stable curves, $r$-spin structures form an étale stack, but both the finiteness and the torsor structure are lost.In the present work, we show how this bad picture can be definitely improved just by placing...

We give a complete classification of stable vector bundles over a cuspidal cubic and calculate their cohomologies. The technique of matrix problems is used, similar to [2, 3].

Per $d\gg g$, vengono trovate curve liscie in ${\mathbb{P}}^3$ di grado $d$ e genere $g$ aventi fibrato normale instabile con grado di instabilità $\sigma$, per ogni $1\le \sigma \le d-4$. Inoltre per $4g-2\le \sigma \le d-4$, viene trovata una famiglia di curve in ${\mathbb{P}}^3$ di grado $d$ e genere $g$ avente fibrato normale instabile con grado di instabilità $\sigma$ e formante uno strato dello schema di Hilbert della giusta dimensione che è $4d-g+1-2\sigma$.

Soit ${\mathrm{𝒮𝒰}}_C\left(r\right)$ l’espace des modules des fibrés vectoriels semi-stables de déterminant trivial sur une courbe lisse $C$ de genre $g\le 2$ sur $C$. On étudie dans cet article, un exemple de fibré introduit par Raynaud dans [4], ne possédant pas de diviseur thêta. On construit ensuite des extensions stables de ce fibré ce qui conduit à une majoration de la codimension du lieu de base du fibré déterminant sur ${\mathrm{𝒮𝒰}}_C\left(r\right)$.

We show that the moduli space of SUX (r, L) of rank r bundles of fixed determinant L on a smooth projective curve X is separably unirational.

This is a slightly expanded version of the talk given by the first author at the conference Instantons in complex geometry, at the Steklov Institute in Moscow. The purpose of this talk was to explain the algebraic results of our paper Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces. In this paper we compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles...

The Gauss-Manin connection for nonabelian cohomology spaces is the isomonodromy flow. We write down explicitly the vector fields of the isomonodromy flow and calculate its induced vector fields on the associated graded space of the nonabelian Hogde filtration. The result turns out to be intimately related to the quadratic part of the Hitchin map.

Let C be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic zero. For a fixed line bundle L on C, let M C (r; L) be the coarse moduli space of semistable vector bundles E over C of rank r with ∧r E = L. We show that the Brauer group of any desingularization of M C(r; L) is trivial.

Let $X$ be a general proper and smooth curve of genus $2$ (resp. of genus $3$) defined over an algebraically closed field of characteristic $p$. When $3\le p\le 7$, the action of Frobenius on rank $2$ semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order $2$ line bundle over $X$. Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp. the abelian...

This paper deals with rank two connections on the projective line having four simple poles with prescribed local exponents 1/4 and $-1/4$. This Lamé family of connections has been extensively studied in the literature. The differential Galois group of a Lamé connection is never maximal : it is either dihedral (finite or infinite) or reducible. We provide an explicit moduli space of those connections having a free underlying vector bundle and compute the algebraic locus of those reducible connections....