Editing Gerbe (section)

==== Moduli stack of stable bundles on a curve ====
Consider a smooth [[projective variety|projective]] [[algebraic curve|curve]] <math>C</math> over <math>k</math> of genus <math>g > 1</math>. Let <math>\mathcal{M}^s_{r, d}</math> be the [[moduli space|moduli stack]] of [[stable vector bundle]]s on <math>C</math> of rank <math>r</math> and degree <math>d</math>. It has a [[Moduli space#Coarse moduli spaces|coarse moduli space]] <math>M^s_{r, d}</math>, which is a [[quasiprojective variety]]. These two moduli problems parametrize the same objects, but the stacky version remembers [[automorphism]]s of vector bundles. For any stable vector bundle <math>E</math> the automorphism group <math>Aut(E)</math> consists only of scalar multiplications, so each point in a moduli stack has a stabilizer isomorphic to <math>\mathbb{G}_m</math>. It turns out that the map <math>\mathcal{M}^s_{r, d} \to M^{s}_{r, d}</math> is indeed a <math>\mathbb{G}_m</math>-gerbe in the sense above.<ref>{{cite journal|last1=Hoffman|first1=Norbert|year=2010|title=Moduli stacks of vector bundles on curves and the King-Schofield rationality proof|journal=Cohomological and Geometric Approaches to Rationality Problems|series=Progress in Mathematics |volume=282 |pages=133–148|doi=10.1007/978-0-8176-4934-0_5|arxiv=math/0511660|isbn=978-0-8176-4933-3|s2cid=5467668}}</ref> It is a trivial gerbe if and only if <math>r</math> and <math>d</math> are [[coprime]].