The Moduli Space of Twisted Exact Differential Forms on Curves in Positive Characteristic

After fixing a pattern $\mathbf{m}$ of zeroes and poles, we introduce a Moduli stack $\mathrm{\Gamma }{\mathcal{M}}_{g,n}^{ex,\mathbf{m}}$ over ${\mathbb{F}}_p$ that parametrizes smooth marked curves together with a non-zero differential form that is the differential of a meromorphic function. Furthermore, we consider the stack ${\mathcal{M}}_{g,n}^{ex,\mathbf{m}}$ that parametrizes those divisors on smooth curves that appear as divisors of exact differential forms. By introducing a local-global principle for first order deformations of the objects that we consider, we show smoothness of these stacks and compute their dimension.