I'm reading Griffiths and Harris, On the Noether-Lefschetz Theorem and Some Remarks on Codimension-two Cycles, Math. Ann. 271, 31-51 (1985).
Their proof of the Noether-Lefschetz theorem is based on degenerating a degree-$d$ surface in $\mathbb{P}^3$ into the union of a plane and a degree-$(d-1)$ surface.
To be specific, we have the following:
- A smooth, degree $(d-1)$ surface $T \subset \mathbb{P}^3$, with $T = V(F)$.
- A plane $P \subset \mathbb{P}^3$, generic w.r.t $T$, with $P = V(L)$.
- A degree $d$ surface $U \subset \mathbb{P}^3$, generic w.r.t. $T$ and $P$, with $U = V(G)$.
We define the family $X \xrightarrow{\pi} \mathbb{P}^1$, where
$$X := \{(x, [s:t]) \in \mathbb{P}^3 \times \mathbb{P}^1 \; | \; s F(x) L(x) + t G(x) = 0\}$$
so that we have the fibers $X_0 \cong T \cup P$, $X_\infty \cong U$. (Here we're talking the affine slice $[1:\ast]$.) We're only concerned with the behavior near zero, so we can restrict $\pi$ and $X$ to a map $X \xrightarrow{\pi} \Delta$, where $\Delta$ is a small disc centered around zero.
Now $X$ will be some singularities with $t = 0$; in particular, we have
$$\frac{\partial}{\partial x_i} (F L - t G) = \frac{\partial F}{\partial x_i} L + \frac{\partial L}{\partial x_i} F - t \frac{\partial G}{\partial x_i}$$ $$\frac{\partial}{\partial t} (F L - t G) = -G$$
So the points with $F = L = G = 0$, $t = 0$ are singularities. Of course, these points are the points of $X_0$ corresponding to $P \cap T \cap U$; assuming nice behavior, we have $d(d-1)$ singular points $\{p_1, \ldots, p_{d(d-1)}\}$ on $t = 0$.
Question: How do I -- rigorously! -- show that these singularities are isolated double points, so that I can in particular take $\Delta$ small enough that these are the only singularities of the (restricted) $X$?
My thoughts: I feel that the point is that I should be able to distort the coordinate system by some analytic function so that, locally at each $p_i$, the equation of $X$ becomes
$$x y - t z = 0.$$
Obviously rotating the coordinates so that $L$ is just $x$ is straightforward, and I guess we could do something like the Weierstrass division theorem to write $F$ as $y$; since we get to choose $P$ generically w.r.t $T$, we can in particular insist that it meets $T$ transversely; this condition is just saying that $P$ isn't the tangent plane to $T$ at $p_i$. Then I have something like $x y - t g(x, y, z) = 0$ locally at $p_i$, which as far as I can tell isn't quite enough, though I'm not really familiar with this sort of thing.
This is probably trivial to people who do SCV on a regular basis, but I'd greatly appreciate a proof sketch here because I'm not really comfortable with this sort of thing yet.