Terracini loci and a codimension one Alexander-Hirschowitz theorem

The Terracini locus T(n, d; x) is the locus of all finite subsets S ⊂ Pⁿ of cardinality x such that S = Pⁿ, h⁰(I2S(d)) > 0, and h¹(I2S(d)) > 0. The celebrated Alexander-Hirschowitz Theorem classifies the triples $(n,d,x)$ for which (n, d; x)=xn. Here we fully characterize the next step in the case $n=2$, namely, we prove that 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≡ 1,2 3 and $x=(d+2)(d+1)/6$.