  In the round 6-sphere, null-torsion holomorphic curves are fundamental
examples of minimal surfaces. This class of minimal surfaces is quite rich: By
a theorem of Bryant, extended by Rowland, every closed Riemann surface may be
conformally embedded in the round 6-sphere as a null-torsion holomorphic curve.
  In this work, we study the second variation of area for compact null-torsion
holomorphic curves $\Sigma$ of genus $g$ and area $4\pi d$, focusing on the
spectrum of the Jacobi operator. We show that if $g \leq 6$, then the
multiplicity of the lowest eigenvalue $\lambda_1 = -2$ is exactly equal to
$4d$. Moreover, for any genus, we show that the nullity is at least $2d + 2 -
2g$. These results are likely to have implications for the deformation theory
of asymptotically conical associative 3-folds in $\mathbb{R}^7$, as studied by
Lotay. Details will appear in an updated version of this report.
