Find stratification to decompose constructible sheaf to constant parts (example from Wikipedia)

Question

I have a question about techniques used in determining the stratification over which a constructible sheaf falls into even constant pieces demonstrated on this example from Wikipedia.

Let $f:X = \text{Proj}(\frac{\mathbb{C}[s,t][x,y,z]}{(s \cdot t \cdot f(x,y,z))}) $ $ \to \text{Spec}(\mathbb{C}[s,t])$

given canonically, where $f(x,y,z)$ homogeneous polynomial determining a smooth plane curve $C_f \subset \mathbb{P}^2$. Then the fibers over closed subscheme $V(s \cdot t)$ are isomorphic to $\mathbb{P}^2$, and over the the complement $\text{Spec}(\mathbb{C}[s,t])- V(s \cdot t)$ are naturally isomorphic to $C_f$.

Consider the derived pushforward $\mathbf{R}f_!(\underline{\mathbb{Q}}_X)$ of the constant sheaf $\underline{\mathbb{Q}}_X$.

Then it is claimed that for restrictions over $V(s \cdot t)$ holds

$\mathbf{R}^kf_!(\underline{\mathbb{Q}}_X)|_{\mathbb{V}(st)} = \underline{\mathbb{Q}}_{\mathbb{V}(st)} \ \ \ ,k=0,2,4 \\ \mathbf{R}^kf_!(\underline{\mathbb{Q}}_X)|_{\mathbb{V}(st)} = \underline{0}_{\mathbb{V}(st)} \ \ \ ,else $

and over complement $\text{Spec}(\mathbb{C}[s,t])- V(s \cdot t)$

$\mathbf{R}^kf_!(\underline{\mathbb{Q}}_X)|_{\text{Spec}(\mathbb{C}[s,t])- V(s \cdot t)} = \underline{\mathbb{Q}}_{\text{Spec}(\mathbb{C}[s,t])- V(s \cdot t)} \ \ \ ,k=0,2 \\ \mathbf{R}^kf_!(\underline{\mathbb{Q}}_X)|_{\text{Spec}(\mathbb{C}[s,t])- V(s \cdot t)} = \underline{\mathbb{Q}}^{\oplus 2g}_{\text{Spec}(\mathbb{C}[s,t])- V(s \cdot t)} \ \ \ ,k=1 \\ \mathbf{R}^kf_!(\underline{\mathbb{Q}}_X)|_{\text{Spec}(\mathbb{C}[s,t])- V(s \cdot t)} = \underline{0}_{\text{Spec}(\mathbb{C}[s,t])- V(s \cdot t)} \ \ \ ,else $

where $g$ the genus of $C_f$.

Question: Could it be elaborated how to obtain this result and which techniques are involved?

Since $f$ is proper, the fibers are easy to determine in light of proper base change: $(\mathbf{R}^kf_!(\underline{\mathbb{Q}}_X))_y= H^i(X_y, \underline{\mathbb{Q}}_{X_y})$.

But even if this implies that on $\text{Spec}(\mathbb{C}[s,t])- V(s \cdot t)$ and $V(s \cdot t)$ the fibers not change, this a priori not implies that restricted to these two pieces the pushforward sheaves are constant there. Naively, they could be also only locally constant there.

Why is this here not the case, and the stratification by $V(s \cdot t)$ decomposes the $\mathbf{R}^kf_!(\underline{\mathbb{Q}}_X)$ already into constant sheaves there?

Answer 1

I'm going to use $g$ for the equation of the curve since using $f$ for both the equation of the curve and the map to $\operatorname{Spec}(\mathbb C[s,t])$ leads to ambiguity.

The situation just using the stalk form of proper base change is worse than you say, because the pushforward sheaves need not be locally constant just given the information on stalks. Instead you could have sheaves like $i_* \mathbb Q \oplus j_! \mathbb Q$ where $i$ is a closed immersion and $j$ is the complementary open immersion.

Here it's very convenient to use the general form of proper base change, i.e. the pullback of $R^k f_!$ under an arbitrary map is equal to the derived pushforward along the base change of $f$ by that map.

Base-changing along the closed immersion of $V(s\cdot t)$, $f$ becomes

$$ \text{Proj}(\frac{\mathbb{C}[s,t][x,y,z]}{(s \cdot t )}) \to \text{Spec}(\mathbb{C}[s,t]/ (s\cdot t))$$

which is the projection $\mathbb P^2 \times \text{Spec}(\mathbb{C}[s,t]/ (s\cdot t))$, i.e. the base change of $\mathbb P^2 \to \textrm{pt}$ by $\text{Spec}(\mathbb{C}[s,t]/ (s\cdot t))\to \textrm{pt}$. Applying the proper base change theorem to that base change we get that the restrictions of $R^k f_! \mathbb Q$ are all pullbacks from a point, i.e. constant sheaves.

Similarly, the base change along the complementary open immersion is

$$ \text{Proj}(\frac{\mathbb{C}[s,t, s^{-1}, t^{-1} ][x,y,z]}{(s \cdot t \cdot g(x,y,z) )}) \to \text{Spec}(\mathbb{C}[s,t, s^{-1}, t^{-1} ]$$

or equivalently

$$ \text{Proj}(\frac{\mathbb{C}[s,t, s^{-1}, t^{-1} ][x,y,z]}{( g(x,y,z) )}) \to \text{Spec}(\mathbb{C}[s,t, s^{-1}, t^{-1} ]$$

which is product morphism $C_g \times \text{Spec}(\mathbb{C}[s,t, s^{-1}, t^{-1} ] \to \text{Spec}(\mathbb{C}[s,t, s^{-1}, t^{-1} ]$, i.e. the base change of $C_g \to \textrm{pt}$ along $\text{Spec}(\mathbb{C}[s,t, s^{-1}, t^{-1} ]\to \textrm{pt}$.

Again we get a constant sheaf for this reason.