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 \(\langle S \rangle = \mathbb {P}^{n}\) , \(h^{0}(\mathcal {I}_{2S}(d)) > 0\) , 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 \((d,x)\in \{(4,4),(4,6),(5,6),(5,7),(6,9),(6,10)\}\) , or \(d\ge 7\) , \(d\equiv 1,2 \pmod {3}\) and \(x=(d+2)(d+1)/6\) .
