Numerical uniruledness from positive anticanonical degree

Investigate whether positivity of the anticanonical degree along a single nonconstant map from a smooth projective curve, namely -K_X\cdot f_*[C]>0, can by itself produce a rational curve through disk continuation under a strict area bound.

Background

The paper's analytic descent argument relies on a global positive Ricci lower bound to decrease the area of holomorphic disks above an explicit threshold. The authors ask whether this global curvature hypothesis can be replaced by the numerical condition that a single curve has positive anticanonical degree.

The question is motivated by the Miyaoka–Mori numerical criterion for uniruledness. Proposition 6.1 treats only the strict subrange $-K_X\cdot f_*[C]>n(g-1)$, while the general criterion requires only positivity of the anticanonical degree. Resolving the problem would require an area-decreasing estimate for almost-minimizing disks under the numerical hypothesis, together with the compactness needed to continue the disks.