On the Hilbert functions of the elements of the Terracini loci of the Veronese varieties

We study the Hilbert functions (often the most extreme ones) of the finite subsets \(S\subset {\mathbb {P}}^n\) $S\subset {\mathbb{P}}^n$ which are Terracini for the order d Veronese embedding of \({\mathbb {P}}^n\) ${\mathbb{P}}^n$ , i.e. S spans \({\mathbb {P}}^n\) ${\mathbb{P}}^n$ , the fat scheme \(2S:= \cup _{p\in S}2p\) $2S:={\cup }_{p\in S}2p$ is contained in a degree d hypersurface and 2S is defective in degree d. Call \({\mathbb {T}}(n,d;x)\) $\mathbb{T}(n,d;x)$ the set of all such sets S with \(\#S=x\) $\#S=x$ . We compute or bound the minimum and the maximum of the first degree of a hypersurface containing S (resp. 2S) and the index of regularity of S (resp. 2S) when S varies in \({\mathbb {T}}(n,d;x)\) $\mathbb{T}(n,d;x)$ . We give stronger results on the Hilbert function of S and 2S for S in some defined subset of \({\mathbb {T}}(n,d;x)\) $\mathbb{T}(n,d;x)$ .