Eisenbud–Harris conjecture on curves of large genus in projective space

Let $C$ be a reduced, irreducible, nondegenerate curve of genus $g$ and degree $d$ in ${\bf P}^n$, and let $\pi_\alpha(d,n)$ denote the Eisenbud–Harris bound, with $\alpha$ in the range under consideration. Eisenbud–Harris conjecture. If

$$d \geq 2n+2\alpha-1$$

and $g>\pi_\alpha(d,n)$, then $C$ lies on a surface of degree at most $n+\alpha-2$. The conjecture sharpens the known theorem by replacing the exponential degree threshold $d_0(n)$ with the stated linear bound; the cases $\alpha=0$ and $\alpha=1$ are known, while the paper establishes the case $\alpha=2$ for $n\geq 8$ and partial results for $\alpha=3,4$.