Control of global sections on the canonical extension

Determine how to control the rational functions on the canonical extension Z_M arising from global sections of tautological symmetric powers of the canonical extension V^\omega, in order to use those functions to investigate the function theory of Z_M.

Background

The paper observes that the canonical extension V\omega has abundant global sections of its tautological symmetric powers. These sections induce rational functions on Z_M, but the authors do not obtain an abstract method for controlling their properties.

The unresolved issue is how to exploit these sections to study the holomorphic or meromorphic function theory of Z_M. The paper instead constructs a holomorphic function directly on Z_M in the specific ruled-surface setting.

References

In summary we obtain an abundance of global sections of $_{(V\omega)}(k)$ defining rational functions on $Z_M$, but it is not clear how to control their properties in an abstract setting and use them to investigate the function theory of $Z_M$.

— Canonical extensions of ruled surfaces over elliptic curves  (2610.03048 - Denisi et al., 2 Oct 2026) in Section 1.2 (Strategy of proof)