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 $0<i<2p+1$ 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 $1<i<2p+1$ 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.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.