2021/07/28 by Beheshti, Roya, Riedl, Eric · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2107.13584
We study rational surfaces on very general Fano hypersurfaces in ℙn, with an eye toward unirationality. We prove that given any fixed family of rational surfaces, a very general hypersurface of degree d sufficiently close to n and n sufficiently large will admit no maps from surfaces in that family. In particular, this shows that for such hypersurfaces, any rational curve in the space of rational curves must meet the boundary. We also prove that for any fixed ratio α, a very general hypersurface in ℙn of degree d sufficiently close to n will admit no maps from a surface satisfying H2 ≥ αHK, where H is the pullback of the hyperplane class from ℙn and K is the canonical bundle on the surface.