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Let $S$ be a complete intersection surface defined by a net $\Lambda$ ofquadrics in $\mathbb P^5$. In this paper we analyze GIT stability of nets ofquadrics in $\mathbb P^5$ up to projective equivalence, and discuss someconnections between a net of quadrics and the associated discriminant sexticcurve. In particular, we prove that if $S$ is normal and the discriminant$\Delta(S)$ of $S$ is stable then $\Lambda$ is stable. And we prove that if $S$has the reduced discriminant and $\Delta(S)$ is stable then $\Lambda$ isstable. Moreover, we prove that if $S$ has simple singularities then$\Delta(S)$ has simple singularities.

Stability of nets of quadrics in $\mathbb{P}^5$ and associated discriminants