Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of SL in PicSL, or a form of it containing the Néron–Severi torus. Let G be the G-torsor over SL obtained by extension of structure group from a universal torsor T over SL. We prove that G does not descend to S unless T does. This is in contrast to a result of Friedman and Morgan that such G always descend to singular del Pezzo surfaces over C from their desingularizations.

A missing hypothesis is added to Lemma 2.2.1 in [Doc. Math. 15 (2010), 35-72]. The rest of the text remains unchanged.

Let X be an irreducible smooth projective curve, of genus at least two, over an algebraically closed field k. Let denote the moduli stack of principal G-bundles over X of fixed topological type $$d \in \pi _1(G)$$ , where G is any almost simple affine algebraic group over k. We prove that the universal bundle over is stable with respect to any polarization on . A similar result is proved for the Poincaré adjoint bundle over $$X \,{\times }\, M_G^{d, {\mathrm {rs}}}$$ , where $$M_G^{d, {\mathrm {rs}}}$$ is the coarse moduli space of regularly stable principal G-bundles over X of fixed topological type d.

Let $M_{g, n}$ (respectively, $\overline{M_{g, n}}$) be the moduli space of smooth (respectively stable) curves of genus $g$ with $n$ marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to determine whether or not $M_{g, n}$ (or equivalently, $\overline{M_{g, n}}$) is a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G. Farkas assert that $M_{g, n}$ is not unirational for any $n \geqslant 0$ if $g \geqslant 22$. Moreover, P. Belorousski and A. Logan showed that $M_{g, n}$ is unirational for only finitely many pairs $(g, n)$ with $g \geqslant 1$. Finding the precise range of pairs $(g, n)$, where $M_{g, n}$ is rational, stably rational or unirational, is a problem of ongoing interest. In this paper we address the rationality problem for twisted forms of $\overline{M_{g, n}}$ defined over an arbitrary field $F$ of characteristic $\neq 2$. We show that all $F$-forms of $\overline{M_{g, n}}$ are stably rational for $g = 1$ and $3 \leqslant n \leqslant 4$, $g = 2$ and $2 \leqslant n \leqslant 3$, $g = 3$ and $1 \leqslant n \leqslant 14$, $g = 4$ and $1 \leqslant n \leqslant 9$, $g = 5$ and $1 \leqslant n \leqslant 12$.

LetSbe a split family of del Pezzo surfaces over a discrete valuation ring such that the general fiber is smooth and the special fiber hasADE-singularities. LetGbe the reductive group given by the root system of these singularities. We construct aG-torsor overSwhose restriction to the generic fiber is the extension of structure group of the universal torsor. This extends a construction of Friedman and Morgan for individual singular del Pezzo surfaces. In case of very good residue characteristic, this torsor is unique and infinitesimally rigid.

Let $\text{M}_C( 2, \mathcal{O}_C) \cong \mathbb{P}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $\mathbb{C}$. Let $\beta_C$ denote the natural Brauer class over the stable locus. We prove that if $f^*( \beta_{C'}) = \beta_C$ for some birational map $f$ from $\text{M}_C( 2, \mathcal{O}_C)$ to $\text{M}_{C'}( 2, \mathcal{O}_{C'})$, then the Jacobians of $C$ and of $C'$ are isomorphic as abelian varieties. If moreover these Jacobians do not admit real multiplication, then the curves $C$ and $C'$ are isomorphic. Similar statements hold for Kummer surfaces in $\mathbb{P}^3$ and for quadratic line complexes.

We characterize all fields of definition for a given coherent sheaf over a projective scheme in terms of projective modules over a finite-dimensional endomorphism algebra. This yields general results on the essential dimension of such sheaves. Applying them to vector bundles over a smooth projective curve C, we obtain an upper bound for the essential dimension of their moduli stack. The upper bound is sharp if the conjecture of Colliot-Th\'el\`ene, Karpenko and Merkurjev holds. We find that the genericity property proved for Deligne-Mumford stacks by Brosnan, Reichstein and Vistoli still holds for this Artin stack, unless the curve C is elliptic.

Let A be a del Pezzo order on the projective plane over the field of complex numbers. We prove that every torsion-free A-module of rank one can be deformed into a locally free A-module of rank one.

In this paper we present a construction of stable bundles on Calabi-Yau threefolds using the method of bundle extensions. This construction applies to any given Calabi-Yau threefold with h^{1,1}>1. We give examples of stable bundles of rank 2 and 4 constructed out of pure geometric data of the given Calabi-Yau space. As an application, we find that some of these bundles satisfy the physical constraint imposed by heterotic string anomaly cancellation.

Let G be a reductive group over an algebraically closed field k . Consider the moduli space of stable principal G -bundles on a smooth projective curve C over k . We give necessary and sufficient conditions for the existence of Poincaré bundles over open subsets of this moduli space, and compute the orders of the corresponding obstruction classes. This generalizes the previous results of Newstead, Ramanan and Balaji–Biswas–Nagaraj–Newstead to all reductive groups, to all topological types of bundles, and also to all characteristics.

Let X and X' be compact Riemann surfaces of genus at least 3, and let G and G' be nonabelian reductive complex groups. If one component M_G^d(X) of the moduli space for semistable principal G-bundles over X is isomorphic to another component M_G'^d'(X'), then X is isomorphic to X'.

We construct a class of stable SU(5) bundles on an elliptically fibered Calabi-Yau threefold with two sections, a variant of the ordinary Weierstrass fibration, which admits a free involution. The bundles are invariant under the involution, solve the topological constraint imposed by the heterotic anomaly equation and give three generations of Standard Model fermions after symmetry breaking by Wilson lines of the intermediate SU(5) GUT-group to the Standard Model gauge group. Among the solutions we find some which can be perturbed to solutions of the Strominger system. Thus these solutions provide a step toward the construction of phenomenologically realistic heterotic flux compactifications via non-Kahler deformations of Calabi-Yau geometries with bundles. This particular class of solutions involves a rank two hidden sector bundle and does not require background fivebranes for anomaly cancellation.

Let C be a smooth projective curve over an algebraically closed field of arbitrary characteristic. Let M r,L ss denote the projective coarse moduli scheme of semistable rank r vector bundles over C with fixed determinant L. We prove Pic(M r,L ss ) = ℤ, identify the ample generator, and deduce that M r,L ss is locally factorial. In characteristic zero, this has already been proved by Drézet and Narasimhan. The main point of the present note is to circumvent the usual problems with Geometric Invariant Theory in positive characteristic.

Motivated by Yang-Mills theory in 4n dimensions, and generalizing the notion due to Atiyah, Drinfeld, Hitchin and Manin for n=1, Okonek, Spindler and Trautmann introduced instanton bundles and special instanton bundles as certain algebraic vector bundles of rank 2n on the complex projective space P^{2n+1}. The moduli space of special instanton bundles is shown to be rational.

LetXXbe a geometrically connected smooth projective curve of genusgX≥2g_X \geq 2overR\mathbb {R}. LetM(r,ξ)M(r, \xi )be the coarse moduli space of geometrically stable vector bundlesEEoverXXof rankrrand determinantξ\xi, whereξ\xiis a real point of the Picard varietyPic_d(X)\underline {\mathrm {Pic}}^d( X). IfgX=r=2g_X = r = 2, then letddbe odd. We compute the Brauer group ofM(r,ξ)M(r,\xi ).

Let C be a smooth projective curve over an algebraically closed field k of arbitrary characteristic. Given a linear algebraic group G over k, let M G be the moduli stack of principal G-bundles on C. We determine the set of connected components π0(MG) for smooth connected groups G.

This mainly expository text translates into stack language the proof of King and Schofield for the rationality of moduli schemes of vector bundles on a curve in the coprime case. An appendix summarizes some basic properties of the relevant moduli stacks.

Let G be an affine reductive algebraic group over an algebraically closed field k . We determine the Picard group of the moduli stacks of principal G -bundles on any smooth projective curve over k .