# Higher cohomology of projective bundles

Let $$C$$ be a curve and $$L$$ be a line bundle with sufficiently large degree. Let $$C_p$$ denote the $$p$$-th symmetric product of $$C$$, which consists of all the effective divisors of degree $$p$$ on $$C$$. Let $$pr:C\times C_p\to C$$ be the projection and $$\sigma:C\times C_p\to C_{p+1}$$ defined by $$(x,\xi_p)\mapsto x+\xi_p$$. Let $$E_L$$ be the locally free sheaf $$\sigma_*pr^*\mathcal{O}(L)$$ on $$C_{p+1}$$. Let $$B$$ be the projective bundle $$\mathbb{P}(E_L)$$ with the natural projection $$\pi:B\to C_{p+1}$$. Is it true that $$H^i(B,\mathcal{O}(d))=0$$ for all $$0 and $$d$$ an arbitrary integer?

By some results by Ein and others(see Singularities and syzygies of secant varieties of nonsingular projective curves), $$\mathcal{O}(1)$$ maps $$B$$ to the $$p$$-th secant variety $$\Sigma_p$$ of $$C$$ in $$\mathbb{P}H^0(C,L)$$ and we have $$H^i(\Sigma_p,\mathcal{O}(d))=0$$ for all $$1 and integers $$d$$. I think if we can further generalize the vanishing result to $$B$$, it helps to understand the syzygies of secant varieties of curves.