A short way of counting maps to hypersurfaces in Grassmannians

Using a Quot scheme compactification, we calculate the virtual count of maps of degree $d$ from a smooth projective curve of genus $g$ to a hypersurface in a Grassmannian, sending specified points of the curve to special Schubert subvarieties restricted to the hypersurface. We study the question of whether this virtual count is in fact enumerative under suitable conditions on the hypersurface, in the regime when the map degree $d$ is large.