Generalization of the separating-morphism obstruction to broader surfaces

Develop the stated lower-bound strategy for separating morphisms on a broader class of smooth real algebraic surfaces satisfying \(\chi(\mathcal O_X)=1\), \(H^2(X,\mathcal L(D_0))=0\), \(\ell+k-\varepsilon<D_0^2-K_X\cdot D_0\), and \(\ell_c-\ell_n-3k+\varepsilon>D_0^2+K_X\cdot D_0+2=2g(D_0)\).

Background

The paper proves non-existence results for separating morphisms of prescribed degree when the curve is embedded in either the projective plane or a Hirzebruch surface. The proofs use a lower bound for the dimension of a linear system, followed by an analysis of pencils and common components to obtain intersection-theoretic inequalities.

The authors then formulate conditions under which a similar argument might apply to other surfaces: $\chi(\mathcal O_X)=1$ and $H^2(X,\mathcal L(D_0))=0$ allow Riemann–Roch to control $\dim|D_0|$, while the two displayed inequalities are intended to yield the required contradiction. They explicitly state that they cannot currently execute the strategy in this more general setting.