Sheaves on Stacky Curves

We develop the basic theory of stacky curves and their categories of sheaves, giving structure results for stacky curves and relating their sheaves to parabolic bundles. We classify the spherical stacky curves and construct a root system inside their Grothendieck Group. We thoroughly describe the geometry of the moduli stack of coherent sheaves on a stacky curve, resulting in several motivic statements, including a formula for the motive of the stack of vector bundles on a stacky curve. Finally we construct a projective good moduli space for the stack of semistable vector bundles on a stacky curve, by constructing determinantal line bundles with many sections on the moduli stack. There is an appendix generalizing global constructions for flattening stratifications to cyclotomic stacks and an appendix collecting techniques for the computation of motives.