Ein–Lazarsfeld conjecture for linear syzygies

Let $\mathbf P^n$ be projective $n$-space, embedded by $\mathcal{O}_{\mathbf P^n}(d)$, and let $\mathrm{K}_{p,1}(\mathbf P^n,\mathcal{O}_{\mathbf P^n}(d))$ denote the Koszul cohomology group governing linear syzygies. Ein–Lazarsfeld conjecture for linear syzygies. Fix $n\geq 2$. For $d\geq n+1$,

$$\mathrm{K}_{p,1}(\mathbf P^n,\mathcal{O}_{\mathbf P^n}(d))=0$$

if $p\geq \binom{d+n-1}{n}+n-1$. This is the $q=1$ vanishing predicted by the broader Ein–Lazarsfeld conjecture; the paper studies this case, while the full conjecture remains open in general.