Ottaviani–Paoletti conjecture for syzygies of Veronese embeddings

Let $k$ be a field of characteristic $0$, let $V$ be a finite-dimensional $k$-vector space, and let $K_{p,q}^d(V,b)$ denote the Koszul cohomology group associated with the $d$-th Veronese embedding, as defined by the three-term complex in the source. The group $K_{p,q}^d(V,0)$ measures minimal $p$-th syzygies of degree $p+q$ of the Veronese coordinate ring.

Ottaviani–Paoletti conjecture.

$$K_{p,q}^d(V,0)=0$$

for $q\geq 2$ and $p\leq 3d-3$.

This is equivalent to the assertion that the $d$-th Veronese embedding satisfies the corresponding linear-syzygy property through this range. The conjecture is known for $d=2$ and when $\dim V=2$ or $3$; the remaining cases were open in the source. Ottaviani and Paoletti showed that the bound is sharp by proving $K_{p,2}^d(V,0)\neq 0$ for $p=3d-2$ when $\dim V\geq 3$ and $d\geq 3$.