Hartshorne–Hirschowitz conjecture for the maximal genus of space curves

For integers $kb5$ and $f\a0\in [k-1,2k-6]$, define

$$A(k,f)=\frac{1}{3}\bigl(k^2-kf+f^2-2k+7f+12+\delta(2k-f-6)\bigr),$$ $$B(k,f)=\frac{1}{3}\bigl(k^2-kf+f^2+6f+11+\delta(2k-f-7)\bigr),$$

where $\delta(c)=3$ if $c=1$ or $3$, $\delta(c)=1$ if $c\equiv2\pmod 3$, and $\delta(c)=0$ otherwise. Let $G(d,k)$ be the maximal genus of an integral curve of degree $d$ in $\mathbb P^3$ not contained in a surface of degree smaller than $k$. Hartshorne–Hirschowitz conjecture. If $d,k>0$ and $A(k,f)\le d<A(k,f+1)$ for some $f\in[k-1,2k-6]$, then

$$G(d,k)=d(k-1)+1-\binom{k+2}{3}+\binom{f-k+4}{3}+h(d),$$

where

$$h(d)=\begin{cases}0&\text{if }A(k,f)\le d\le B(k,f),\\ \frac12(d-B(k,f))(d-B(k,f)+1)&\text{if }B(k,f)\le d<A(k,f+1).\end{cases}$$

This gives the conjectural maximal genus in the range where the general Hartshorne–Hirschowitz formula is not yet known; the paper develops techniques toward proving it, but the statement remains open in the indicated range.