Cohomology bounds for sheaves of dimension one

We find the sharp bounds on $h^0(F)$ for one-dimensional semistable sheaves $F$ on a projective variety $X$ by using the spectrum of semistable sheaves. The result generalizes the Clifford theorem. When $X$ is the projective plane $\mathbb{P}^2$, we study the stratification of the moduli space by the spectrum of sheaves. We show that the deepest stratum is isomorphic to a subscheme of a relative Hilbert scheme. This provides an example of a family of semistable sheaves having the biggest dimensional global section space.