Universal parabolic moduli over \(\overline{M}_{g,n}\) (Q6050192)

For integers \(g \ge 2\) and \(n \ge 1\) let \({\overline M}_{ g,n}\) be the Deligne-Mumford moduli space of marked stable curves of genus \(g\) and number of marked points \(n\). In [\textit{D. Schlüter}, Universal moduli of parabolic sheaves on stable marked curves. Oxford: Oxford University (PhD Thesis) (2011)] the existence of a universal moduli space \(\overline{\mathcal U}_{ g,n,r}\) of parabolic pure sheaves over \({\overline M}_{ g,n}\) is proven. In the treatement of Schlüter, following [\textit{R. Pandharipande}, J. Am. Math. Soc. 9, No. 2, 425--471 (1996; Zbl 0886.14002)], the fiber of the moduli space \(\overline{\mathcal U}_{ g,n,r}\) over \([C ' ] \in {\overline M}_{ g,n}\) is the moduli space of ``\(p_2\)-stable parabolic pure sheaves on \(C'\) modulo \(\hbox{Aut}(C', x_1 ,\dots, x_n )\)''. The object of this paper is the construction of a moduli space which ``parameterizes stable parabolic Gieseker vector bundles''. This space is a normal ``projective variety \(\overline{\mathfrak U}_{g,n,r}\) over \({\overline M}_{ g,n}\) such that the fiber over a marked stable curve \(C'\) parameterizes aut-equivalance classes of pairs \((C, E_* )\) where \(C\) is a marked semi-stable curve whose fixed marked stable model is \(C'\) and \(E_*\) is a stable parabolic Gieseker bundle of fixed numerical type on \(C\)'' (partial quotation of \textbf{Theorem 1.4.}). Moreover, the dimension of \(\overline{\mathfrak U}_{g,n,r}\) is determined and it is shown that it has ``good singularities'' (the precise meaning is given in \textbf{Theorems 1.5.} and \textbf{1.6.}). The paper is made more readable by the material presented in appendices A--C, where indications to the existing literature is given.