Geometric description of Stein factorization in a special case

Question

Let $X\subset \mathbb P^{n+1}_{\mathbb C}$ be a hypersurface of degree $3\leq d\leq n+2$. We have a natural morphism $\varphi:Hilb_2(X)=X^{[2]}\rightarrow G(2,n+2)$, from which we obtain $\pi:\mathbb P(\mathcal E)\rightarrow X^{[2]}$ and $f:\mathbb P(\mathcal E)\rightarrow \mathbb P^{n+1}$ where $\mathcal E$ is the pullbback by $\varphi$ of the rank $2$ quotient bundle on $G(2, n+2)$.
Let $R\subset \mathbb P(\mathcal E)$ be the divisor whose fiber over a general length $2$ subscheme $x_1+x_2$ of $X$ consists in the remaining $d-2$ points of $X$ lying on the line generated by $x_1$ and $x_2$. The morphism $\pi_{|R}:R\rightarrow X^{[2]}$ has degree $d-2$ and the positive dimensional fibers are the lines contained in $X$.

Is there a explicit (geometric) description of the Stein factorization of $\pi_{|R}$?

Answer 1

I am just expanding my comment from above. First consider the problem of "describing" the Stein factorization for the universal degree $e$ divisor on $\mathbb{P}^1_k$. Let $e\geq 1$ be an integer. Denote by $V$ the $2$-dimensional $k$-vector space $H^0(\mathbb{P}^1,\mathcal{O}(1))$, so that $H^0(\mathbb{P}^1,\mathcal{O}(e))$ equals $\text{Sym}^e(V)$. Denote by $\mathbb{A}_e$ the $e$-dimensional affine space with specified linear structure whose associated $k$-vector space of linear functional is identified with $\text{Sym}^e(V)^\vee$ having dimension $e+1$. Denote by $R$ the $k$-algebra $\mathcal{O}_{\mathbb{A}}(\mathbb{A})$, i.e., the polynomial $k$-algebra on the $(e+1)$-dimensional $k$-vector space $\text{Sym}^e(V)^\vee$. This is naturally a graded $k$-algebra, with $R_m = \text{Sym}^m(\text{Sym}^e(V)^\vee)$. The algebra has a natural action of $\textbf{GL}(V)$ that is compatible with the grading.

On the product $\mathbb{A}_e\times_k \mathbb{P}^1_k$, there is a universal sheaf homomorphism, $$s:\text{pr}_{\mathbb{P}^1}^*\mathcal{O}(-e) \to \mathcal{O}_{\mathbb{A}\times \mathbb{P}^1}.$$ Denote by $$i:\mathcal{D}\hookrightarrow \mathbb{A}_e\times_k \mathbb{P}^1_k,$$ the zero scheme of $s$, i.e., the ideal sheaf of $\mathcal{D}$ equals the image of $s$. Thus, there is a short exact sequence, $$0\to \text{pr}_{\mathbb{P}^1}^*\mathcal{O}(-e) \xrightarrow{s} \mathcal{O}_{\mathbb{A}\times \mathbb{P}^1} \to i_*\mathcal{O}_{\mathcal{D}} \to 0.$$ The associated long exact sequence of $\text{pr}_{\mathbb{A}}$ is, $$0 \to R \to \text{pr}_{\mathbb{A},*} \mathcal{O}_{\mathcal{D}} \to R\otimes_k H^1(\mathbb{P}^1,\mathcal{O}(-e)) \to 0.$$ The middle term of this sequence, call it $S$, is the algebra we would like to describe. The first map is the pullback map. With respect to this map, $S$ is a finite, free $R$-module. The trace map of $R\to S$ is a splitting of the short exact sequence as a sequence of $R$-modules. The natural scaling action of $\mathbb{G}_m$ on $\mathbb{A}_e$, $(\lambda,a)\mapsto \lambda\cdot a$, induces a "modified action" of $\mathbb{G}_m$ by $\lambda\bullet a = \lambda^e \cdot a$. This modified action lifts to an action of $\mathbb{G}_m$ on $\mathcal{D}$. In particular, the action of $\mu_e \subset \mathbb{G}_m$ on $S$ has invariants $R$, and $H^1(\mathbb{P}^1,\mathcal{O}(-e))$ seems to, roughly, be playing the role of the quotient of the group algebra $k[\mu_e]$ by the constants, $k\cdot 1$.

I am not sure how explicitly we can "describe" $S$. For each nonzero $x\in V$, for the associated linear functional $x^e$ on $\mathbb{A}_e$, over the basic open affine $D(x^e)$ we can easily write down a free $R[1/x^e]$-basis for $S[1/x^e]$ together with its algebra structure. However, this is not $\textbf{GL}(V)$-invariant (it does have an explicit induced action of the stabilizer subgroup of $x$). More importantly, based on your question you seem to be particularly interested in the fiber of $S$ over the origin, i.e., $S/\mathfrak{m}S$, where $\mathfrak{m}\subset R$ is the maximal ideal of the origin. By restricting the flat $R$-algebra $S$ over the coordinate algebra $R/J$ of the closed subscheme of $\mathbb{A}_e$ parameterizing polynomials with a root of multiplicity $\geq e$, it appears that $S/\mathfrak{m}S$ equals $k[t]/\langle t^e \rangle$ with trivial $\textbf{GL}(V)$-action. It appears that the $\mu_e$-action on $S/\mathfrak{m}S$ induced from the "modified action" is nontrivial, with the action on the subspace $k\cdot \overline{t}$ being the standard action of $\mu_e$. As I mentioned, I do not see any "natural" basis for the algebra that simultenously makes all of the structures evident (the grading, the action of $\textbf{GL}(V)$, and the modified action of $\mu_e$).

To compute the Stein factorization of $\pi|_R$, you can locally trivialize $\mathcal{E}$ and use the induced morphism from $f$ to $\mathbb{A}_{d-2}$. The pullback of $S$ by $f$ is the algebra whose (relative) Spec gives the Stein factorization. Notice, $f$ is (locally) a smooth fiber bundle over $\mathbb{A}_{d-2}$. So if you are interested in properties of the Stein factorization that are smooth local (or fppf local), you may as well work with $S$ over $R$. This algebra is simpler, and it has the additional structures: grading, action of $\textbf{GL}(V)$, etc.