  The Terracini locus $\mathbb{T}(n, d; x)$ is the locus of all finite subsets
$S \subset \mathbb{P}^n$ of cardinality $x$ such that $\langle S \rangle =
\mathbb{P}^n$, $h^0(\mathcal{I}_{2S}(d)) > 0$, and $h^1(\mathcal{I}_{2S}(d)) >
0$. The celebrated Alexander-Hirschowitz Theorem classifies the triples
$(n,d,x)$ for which $\dim\mathbb{T}(n, d; x)=xn$. Here we fully characterize
the next step in the case $n=2$, namely, we prove that $\mathbb{T}(2,d;x)$ has
at least one irreducible component of dimension $2x-1$ if and only if either
$(d,x)=(6,10)$ or $(d,x)=(4,4)$ or $d\equiv 1,2 \pmod{3}$ and $x=(d+2)(d+1)/6$.
