Singularities of the intersection of quasiprojective schemes

Question

I'm getting hung up on what should be a fairly simple application of definitions in Poonen's paper on Bertini over finite fields. Here's the setup:

• $S=\mathbb{F}_q[x_0,\dots,x_n]$
• $S_d$ consists of the degree $d$ homogeneous polynomials in $S$
• $H_f=\operatorname{Proj}(S/(f))$ for some $f\in S_d$
• $X$ is a smooth quasiprojective subscheme of $\mathbb{P}_{\mathbb{F}_q}^n$ for some $n$
• $\dim X=m$
• All intersections are scheme-theoretic

Poonen states that for a closed point $P\in X$,

The condition that $P$ be singular on $H_f\cap X$ amounts to $m+1$ linear conditions on the Taylor coefficients of a dehomogenization of $f$ at $P$, and these linear conditions are over the residue field of $P$.

Assuming this, I understand why we expect the "probability" that $P$ is a singular point of $H_f\cap X$ is $q^{-(m+1)\deg P}$ where $\deg P=[\kappa(P):\mathbb{F}_q]$.

My question is, how does one concretely see these linear conditions? We're picking a closed $P\in X$ and going through the $f\in S_d$, seeing which $H_f\cap X$ are singular at $P$. Let $U$ be an affine open of $X$ containing $P$ with affine coordinates $x_1,\dots,x_n$, and let $\overline{f}$ be the dehomogenization of $f$ in these coordinates. If we were just checking singularity of $H_f$ at $P$, I see $n+1$ conditions: one for the vanishing of $\overline{f}$ and one for the vanishing of each partial $\partial \overline{f}/\partial x_i$. (In other words, the vanishing of the constant and degree 1 Taylor coefficients). This should correspond to the case $H_f=X$, but if $m=\dim H_f=n-1$ (it's a hypersurface), then $m+1=n\neq n+1$.

And in the general case, why does intersecting with $X$ cut down the number of conditions to $m+1$? Somehow the $m+1$ is coming from needing $\dim_{\kappa(P)}\mathfrak{m}_P/\mathfrak{m}_P^2 > \dim\mathcal{O}_{H_f\cap X,P}$. Of course $\dim\mathcal{O}_{X,P}=m$ but what can we say about $\dim\mathcal{O}_{H_f\cap X,P}$? And how can we explicitly realize these $m+1$ conditions in terms of Taylor coefficients?

Any insight into just the case $X=V_+(g_1,\dots,g_r)$, so $H_f\cap X=V_+(f,g_1,\dots,g_r)$, would be enough to determine the general claim, I think.

Answer 1

As $X$ is smooth at $P$, the Jacobian of any dehomogenization of $X$ at $P$ has rank $n-m$. Therefore $H_f\cap X$ is singular at $P$ iff the Jacobian of the dehomogenization of $H_f\cap X$ still has rank $n-m$, i.e. the row vector formed by the partial derivatives of the dehomogenization of $f$ still belongs to the span of the rows from the Jacobian of $X$ at $P$. The condition "belongs to an $n-m$ dimensional subspace in an $n$-dimensional vector space" is equivalent to "has $m$ coordinates zero" when choosing a basis for the $n-m$ dimensional subspace and then extending to a basis for the whole space - now add one linear condition for $H_f$ passing through $P$ and you get $m+1$ linear conditions on the coefficients.

(Your "[t]his should correspond to the case $H_f=X$" is not quite right - the work you do there is really about showing the claim in the case that $X=\Bbb P^n$.)

Answer 2

On an intuitive level, the fact that the dimension of $X$ is $m$ means that $X$ is locally given by $m$ independent variables, so smoothness of a subvariety on $X$ is determined by $m+1$ conditions. When considered inside of $\mathbb P^n$, we have that $X$ will locally be given by $n-m$ equations with linearly independent derivatives, so smoothness of $H_f \cap X$ will be given by $n+1 - (n-m) = m+1$ conditions.

As a toy example, one can consider the case where $X = \mathbb P^m \subset \mathbb P^n$. Then $H_f \cap X$ is the subscheme of $X$ given by the vanishing of the pullback of $f$ to $X$, hence smoothness at a point $P \in X$ depends on $m+1$ conditions. Then, try to see how smoothness of $H_f \cap X$ at $P \in X$ implies smoothness when considered as a subscheme in $\mathbb P^n$.