Terracini loci and a codimension one Alexander–Hirschowitz theorem

Abstract 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 $$x=(d+2)(d+1)/6$$ .