Terracini loci and a codimension one Alexander-Hirschowitz theorem

The Terracini locus $\mathbb{T}(n,d;x)$ is the locus of all finite subsets $S$ of ${\mathbb{P}}^n$ of cardinality $x$ such that $⟨S⟩={\mathbb{P}}^n$, $h^0({\mathcal{I}}_{2S}(d))>0$, 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 $(d,x)\in \{(4,4),(4,6),$ $(5,6),(5,7),$ $(6,9),(6,10)\}$, or $d\ge 7$, $d\equiv 1,2(\mathrm{m}\mathrm{o}\mathrm{d}3)$ and $x=(d+2)(d+1)\mathrm{/}6$.