I've came across this problem while thinking about some properties of fat schemes. Let me give you an explicit (motivating) example:
We have $S=\mathbb{C}[x,y,z]$, the coordinate ring of $\mathbb{P}^2$, and $P=[0,0,1]$. Consider a $5$-symmetric scheme $Z$ concentrated in $P$ (i.e. for every line $L$ passing through $P$ it holds that $\deg(L \cap Z)=5$) of degree $21$ (the degree of a $6-$fat point in $\mathbb{P}^2$). I want to prove that such a scheme $Z$ can't have generic Hilbert function.
If such a scheme indeed had generic Hilbert function then its ideal $I:=I_Z$ would be generated by $7$ sextics, i.e. $I=(F_1,\dots,F_7)$. In order to be $5-$symmetric we must also require that at least two of the $F_i$'s, say $F_1,F_2$, must be written like $F_1= G_1z+H_1$ and $F_2=G_2z+H_2$, with $G_i$ quintics in $\{x,y\}$ without common lines and $H_i$ sextics also in $\{x,y\}$. In order to be concentrated only at $P=[0,0,1]$ we have also to ensure that:
- $\gcd(H_1,H_2,F_3,\dots,F_7)=1$
- $\gcd(G_1+H_1,G_2+H_2,F_3,\dots,F_7)=1$
Making some computations with Macaulay2 I've noticed that every ideal of this form is always not saturate and in the saturation it appears at least a quintic which is in the span of $\langle G_1,G_2 \rangle$.
My question finally is the following: is there a (geometric hopefully) way to prove that this is true?
Let me briefly sketch my idea: we have that $I$ is generated by $I_6$ and the (affine) tangent cone $\overline{I}_5:=TC((I_6)_{|z=1})$ is generated by $(G_1,G_2)$ (or if you prefer the localization of $Z$ at $P$).
Let $$\phi_1:I_6 \times S_k \rightarrow I_{k+6} \subset S_{k+6}$$ $$\phi_2:\overline{I}_5 \times S_{k+1} \rightarrow \overline{I}_{k+6} \subset S_{k+6}$$ with $\phi_2(f,S_{k+1})=\mathbb{C}^{\dim(S_{k+1})} \subset S_{k+6}$ for every $f \in \overline{I}_5$.
This finally induce a map $$\overline{\phi}_2: \overline{I}_5 \rightarrow \operatorname{Gr}(\dim(S_{k+1}),S_{k+6})$$ Under the assumption of my construction of $Z$, is it possible to infer that at least a linear subspace in $\overline{\phi}_2(\overline{I}_5)$ is contained also in $I_{k+6}$? (This would prove that there exist at least a quintic of $\overline{I}_5$ in the saturation of $I$).
I'm not an expert in commutative algebra so maybe I'm not seeing something like a direct algebraic result from which this follows easily. Thanks in advance.
