The pullback of a theta divisor to \(\overline{\mathcal{M}}_{g,n}\) (Q2852456)

Let \(\underline d=(d_1,\dots,d_n)\) be an \(n\)-tuple of integers summing up to \(g-1\) and consider the map NEWLINE\[NEWLINE\phi_{\underline d}: \mathcal M_{g,n}\to \text{Pic}_g^{g-1}NEWLINE\]NEWLINE from the moduli space of smooth curves of genus \(g\) with \(n\) marked points to the universal Picard variety of degree \(g-1\) over \(\mathcal M_g\) given by \(\phi_{\underline d}(C;x_1,\dots,x_n)=\mathcal O_C(d_1x_1+\dots+d_nx_n)\). Let \(\Theta_g\) be the universal theta divisor on Pic\(^{g-1}_g\) parametrizing the locus of effective divisors of degree \(g-1\). Assume that at least one \(d_i\) is negative. Then the image of \(\phi_{\underline d}\) is not contained in \(\Theta_g\). The main result of the present paper is the computation of the class of the pullback of \(\Theta_g\) via \(\phi_{\underline d}\), NEWLINE\[NEWLINED_{\underline d}:=\phi_{\underline d}^*(\Theta_g)=\{[C;x_1,\dots,x_n]\in\mathcal M_{g,n}| h^0(C,d_1x_1+\dots+x_nd_n)\geq 1\}NEWLINE\]NEWLINE and of its closure \(\overline D_{\underline d}\) on the moduli space of stable curves of genus \(g\) with \(n\) marked points \(\overline {\mathcal M}_{g,n}\).NEWLINENEWLINETo find the coefficients of \([\overline D_{\underline d}]\) in the generators of \(\overline{\mathcal M}_{g,n}\) the author starts by expressing \(D_{\underline d}\) as the degeneracy locus of a map of vector bundles. Then he uses a pushdown argument to reduce the number of marked points and then computes several intersections with test curves. Tools from the theory of limit linear series are also extensively used throughout the paper.NEWLINENEWLINEA similar divisor was computed by \textit{R. Hain} in [``Normal functions and the geometry of moduli spaces of curves'', in: G. Farkas (ed.) and I. Morrison (ed.), Handbook of Moduli. Somerville, MA: International Press (2013), \url{arXiv:1102.4031v3}] and both the author's and Hain's results were reproven in a recent work by \textit{S. Grushevsky} and \textit{D. Zakharov} in [``The double ramification cycle and the theta divisor'', Duke Math. J. (to appear), \url{arXiv:1206.7001}].