The author considers the moduli spaces \(\mathcal{V}_{g,\delta,l}\) of irreducible curves with \(\delta\) nodes and \(l\) marked points on a polarised \(K3\) surface of genus \(g\), \(\mathcal{M}_{g,n}\) of curves of genus \(g\) with \(n\) marked points, and \(\mathcal{F}_{g,\delta}\) of polarised \(K3\) surfaces of genus \(g\) with \(\delta\) marked points by studying relations among them.
The first part focuses on the moduli maps \(c_{g,\delta,l}:\mathcal{V}_{g,\delta,l}\to\mathcal{M}_{g-\delta,2\delta+l}/\mathbb{Z}_2^{\oplus\delta}\) and \(\pi:\mathcal{V}_{g,\delta,l}\to\mathcal{F}_{g,\delta+l}\). The main result in this part is to give a criterion on \(g,\,\delta\) and \(l\) that these maps are dominant.
The second part is devoted to the structure of fibres of the map \(c_{g,\delta,l}\), and a condition on \(g\) that the general fibres of the map are irreducible is given. The key to prove the result is to study the map \(c_{11,\delta,0}\).
Finally, it is concluded that the moduli space \(\mathcal{M}_{11,n}\) with \(n\leq6\) is unirational after showing that the space \(\mathcal{F}_{11,n}\) is unirational if \(n\leq6\). It is also obtained that the spaces \(\mathcal{F}_{11,\delta+l}\) and \(\mathcal{M}_{11-\delta,2\delta+l}/\mathbb{Z}_2\) are birationally equivalent in some cases.
The proof is given by standard arguments of the cohomology theory in algebraic geometry. Explicit computations of the dimensions and the Kodaira dimensions are done.