The singular locus of hypersurface sections containing a closed subscheme over finite fields

TL;DR

This paper proves the existence of hypersurfaces over finite fields that contain a given subscheme and intersect a smooth scheme smoothly, providing bounds on singularities and extending Bertini theorems using Poonen's sieve.

Contribution

It extends Bertini theorems over finite fields by establishing conditions for hypersurfaces containing a subscheme to intersect smoothly, with bounds on singular loci and local conditions.

Findings

Existence of hypersurfaces containing a subscheme and intersecting smoothly

Bound on the dimension of the hypersurface's singular locus

Extension of Bertini theorems over finite fields

Abstract

We prove that there exist hypersurfaces that contain a given closed subscheme $Z$ of the projective space over a finite field and intersect a given smooth scheme $X$ off of $Z$ smoothly, if the intersection $V = Z \cap X$ is smooth. Furthermore, we can give a bound on the dimension of the singular locus of the hypersurface section and prescribe finitely many local conditions on the hypersurface. This is an analogue of a Bertini theorem of Bloch over finite fields and is proved using Poonen's closed point sieve. We also show a similar theorem for the case where $V$ is not smooth.

Equations

$${\zeta }_X(s):=\prod _{P\in X\text{ closed}}(1-q^{-s\mathrm{deg}P}{)}^{-1}.$$

$${\mu }_Z(\mathcal{P}):=\underset{d\to \infty }{\lim }\frac{\#(\mathcal{P}\cap I_d)}{\#I_d},$$

$$\mathcal{P}=\{f\in I_{\mathrm{homog}}:$$

$$\displaystyle f\big{|}_{Y}\in T\mbox{ and }\dim(H_{f}\cap U)_{\operatorname{sing}}\leq k\}.$$

$${\mathcal{P}}_r=\{f\in I_{\mathrm{homog}}:$$

$$\displaystyle\mbox{at all points }P\in(U-V)_{<r}\mbox{ and }f\big{|}_{Y}\in T\}$$

$${\mathcal{Q}}_r^{\mathrm{medium}}=\bigcup _{d\ge 0}\{f\in I_d:$$

$$\displaystyle\mbox{such that }H_{f}\cap U\mbox{ is not smooth of dimension $m-1$ at }P\}.$$

$${\mathcal{Q}}_{U-V}^{\mathrm{high}}=\bigcup _{d\ge 0}\{f\in I_d:$$

$$\displaystyle H_{f}\cap U\mbox{ is not smooth of dimension $m-1$ at $P$}\},$$

$${\mathcal{Q}}_V=\{f\in I_{\mathrm{homog}}:\dim ((H_f\cap U{)}_{\mathrm{sing}}\cap V)\ge k+1\}.$$

$$H^{0}(X,\mathcal{I}_{Z}\cdot\operatorname{\mathcal{O}}_{C}(d))=\left\{\begin{array}[]{cl}q^{(m-l)\deg P},&\mbox{ if }P\in V,\\ q^{(m+1)\deg P},&\mbox{ else}.\end{array}\right.$$

$${\mu }_Z({\mathcal{P}}_r)=\frac{\#T}{\#H^0(Y,{\mathcal{I}}_Z\cdot O_Y)}\prod _{P\in (U-V{)}_{<r}}(1-q^{-(m+1)\mathrm{deg}P}{)}^{-1}.$$

$${\mu }_Z({\mathcal{P}}_r)=\frac{\#T}{\#H^0(Y,{\mathcal{I}}_Z\cdot O_Y)}\prod _{P\in (V{)}_{<r}}(1-q^{(m-l)\mathrm{deg}P})\prod _{P\in (U-V{)}_{<r}}(1-q^{-(m+1)\mathrm{deg}P}{)}^{-1}.$$

$${\overline{\mu }}_Z({\mathcal{Q}}_{U-V}^{\mathrm{high}})=0.$$

$${\overline{\mu }}_Z({\mathcal{Q}}_V)=0.$$

$$f=f_0+g_1^p{t}_1+\dots +g_{l-k}^p{t}_{l-k}$$

$$W_i=V\cap \left\{D_{1}f=\dots =D_{i}f=0\right\}.$$

$$\mathcal{P}\subseteq {\mathcal{P}}_r\subseteq \mathcal{P}\cup {\mathcal{Q}}_r^{\mathrm{medium}}\cup {\mathcal{Q}}_{U-V}^{\mathrm{high}}\cap {\mathcal{Q}}_V:$$

$${\mu }_Z(\mathcal{P})=\underset{r\to \infty }{\lim }{\mu }_Z({\mathcal{P}}_r)=\frac{\#T}{\#H^0(Y,{\mathcal{I}}_Z\cdot O_Y)}{\zeta }_{U-V}(m+1{)}^{-1}.$$

$$X_e=X({\Omega }_{X|F_q}^1,e)$$

$$\mathcal{P}=\{f\in I_{\mathrm{homog}}:$$

$$\text{ and such that }\dim (H_f\cap U{)}_{\mathrm{sing}}\le 0\}.$$

$$\mathcal{Q}=\left\{f\in I_{\mathrm{homog}}:\dim (H_f\cap U{)}_{\mathrm{sing}}\ge 1\right\}.$$

$$f=f_0+g_1^p{t}_1+\dots +g_{l_e-1}^p{t}_{l_e}$$

$$W_i=V_e\cap \left\{D_{1}f=\dots =D_{i}f=0\right\},$$

$$\mathcal{P}\subseteq {\mathcal{P}}_r\subseteq \mathcal{P}\cup {\mathcal{Q}}_r^{\mathrm{medium}}\cup {\mathcal{Q}}_{U-V}^{\mathrm{high}}\cap \mathcal{Q}.$$

Full text

1. Introduction

The classical Bertini theorem over infinite fields guarantee the existence of a smooth hypersurface section for a smooth subscheme $X$ of the projective space. For a given closed subvariety $Z\subseteq X$, Bloch showed that the hypersurface can be assumed to contain $Z$, if $V=Z\cap X$ is smooth and $2p>\dim X$, where $1<p$ is the codimension of $Z$ in $X$ ([Blo71]). If the condition on the dimension is not fulfilled, this does not hold anymore. But Bloch showed that there still exists a hypersurface section $Y$ of $X$ such that the singular locus of $Y$ is smooth, contained in $Z$ and of dimension $n-2p$. Over finite fields, Poonen proved an analogue for the case where $V$ is smooth and $2\dim V<\dim X$ ([Poo08]); here $Z$ is a closed subscheme of the projective space. In this paper, we generalize this and show an analogue over finite fields of Bloch’s result for $2\dim V\geq\dim X$, where we also add the possibility to impose finitely many local conditions on the hypersurface (2.1).

Furthermore, we show that if the intersection $V$ is not smooth and $2\dim V=\dim X$, there exists a hypersurface $H$ containing $Z$ and intersecting $X$ off of $Z$ smoothly, such that the singular locus of $H\cap X$ consists at most of finitely many points. We use Poonen’s closed point sieve (cf. [Poo04]) to prove this, and also prescribe finitely many local conditions on the hypersurface (3.1).

We use the following notation: let $\operatorname{\mathbb{F}}_{q}$ be a finite field of $q=p^{a}$ elements. Let $S=\mathbb{F}_{q}\left[x_{0},\ldots,x_{n}\right]$ and let $S_{d}\subseteq S$ the $\mathbb{F}_{q}$-subspace of homogeneous polynomials of degree $d$. Let $S_{\operatorname{homog}}=\bigcup_{d\geq 0}S_{d}$ and let $S_{d}^{\prime}$ be the set of all polynomials in $\operatorname{\mathbb{F}}_{q}[x_{0},\ldots x_{n}]$ of degree $\leq d$.

For a scheme $X$ of finite type over $\operatorname{\mathbb{F}}_{q}$, we define the zeta function of $X$ as

[TABLE]

This product converges for $\mbox{Re}(s)>\dim X$.

Let $Z$ be a fixed closed subscheme of $\operatorname{\mathbb{P}}^{n}=\operatorname{\mathbb{P}}^{n}_{\operatorname{\mathbb{F}}_{q}}$. For $d\in\operatorname{\mathbb{Z}}_{\geq 0}$ let $I_{d}$ be the $\operatorname{\mathbb{F}}_{q}$-subspace of polynomials $f\in S_{d}$ vanishing on $Z$, and $I_{\operatorname{homog}}=\bigcup_{d\geq 0}I_{d}$. For a polynomial $f\in I_{d}$ let $H_{f}=\operatorname{Proj}(S/(f))$ be the hypersurface defined by $f$.

As in Poonen’s paper ([Poo08]), we want to measure the density of a set of polynomials within the space of polynomials vanishing on $Z$, and define the density relative to the closed subscheme $Z$ of a subset $\mathcal{P}\subseteq I_{\operatorname{homog}}$ to be

[TABLE]

if the limit exists.

Acknowledgments. The author would like to thank Uwe Jannsen heartily for his support and help. Furthermore, the author gratefully acknowledges support from the Deutsche Forschungsgemeinschaft through the SFB 1085 Higher Invariants.

2. Results for V smooth

Following [Poo04], we want to prescribe finitely many local conditions on the hypersurface. For this, let $Y$ be a finite subscheme of $\operatorname{\mathbb{P}}^{n}$. For a polynomial $f\in I_{d}$ we define f\big{|}_{Y}\in H^{0}(Y,\mathcal{I}_{Z}\cdot\operatorname{\mathcal{O}}_{Y}) as follows: on each connected component $Y_{i}$ of $Y$ let f\big{|}_{Y} be equal to the restriction of $x_{j}^{-d}f$ to $Y_{i}$, where $j=j(i)$ is the smallest $j\in\left\{0,1,\ldots,n\right\}$ such that the coordinate $x_{j}$ is invertible on $Y_{i}$.

Theorem 2.1**.**

Let $X$ be a quasi-projective subscheme of $\operatorname{\mathbb{P}}^{n}$ and $Y$ a finite subscheme of $\operatorname{\mathbb{P}}^{n}$ such that $U=X-(X\cap Y)$ is smooth of dimension $m\geq 0$ over $\operatorname{\mathbb{F}}_{q}$. Let $Z$ be a closed subscheme of $\operatorname{\mathbb{P}}^{n}$ such that $Z\cap Y=\emptyset$. Let $T\subseteq H^{0}(Y,\mathcal{I}_{Z}\cdot\operatorname{\mathcal{O}}_{Y})$. Assume that the intersection $V:=Z\cap U$ is smooth of dimension $l$ such that $2l\leq m+k$ where $k\in\operatorname{\mathbb{Z}}_{\geq 0}$. Define

[TABLE]

Then $\mu_{Z}(\mathcal{P})=\dfrac{\#T}{\#H^{0}(Y,\mathcal{I}_{Z}\cdot\operatorname{\mathcal{O}}_{Y})}\zeta_{U-V}(m+1)^{-1}$.

In particular, there exists a hypersurface $H$ containing $Z$, defined by a polynomial $f$ such that f\big{|}_{Y}\in T and such that the singular locus of the hypersurface section is contained in $Z$ and at most of dimension $k$. The existence of such a hypersurface has been shown for infinite fields as well ([Blo71], Proposition 1.2).

Remark \therem.

The proof is a closed point sieve as introduced in [Poo04] and is organized as follows: First we look at closed points of low degree, defining the relevant set of polynomials to be

[TABLE]

and calculate the density of $\mathcal{P}_{r}$ in section 2.1. As in [Poo04], the argument does not work if we let $r$ tend to infinity before we measure the density. Hence we also need to consider closed points of medium and high degree. We fix $c$ such that $S_{1}I_{d}=I_{d+1}$ for all $d\geq c$ (cf. [Poo08]). For closed points of medium degree, let

[TABLE]

For closed points of high degree, we differentiate between points on and off of $V$, and define

[TABLE]

and

[TABLE]

We will show in 2.2, 2.3 and 2.3, that the error in the approximation that we make by considering $\mathcal{P}_{r}$ instead of $\mathcal{P}$ tends to zero for $r\rightarrow\infty$, i.e. the sets $Q_{r}^{\operatorname{medium}}$, $Q^{\operatorname{high}}_{U-V}$ and $Q_{V}$ are of density zero for $r\rightarrow\infty$.

2.1. Points of low degree

Lemma \thelemma.

Suppose $\operatorname{\mathfrak{m}}\subseteq\operatorname{\mathcal{O}}_{U}$ is the ideal sheaf of a closed point $P\in U$. Let $C\subseteq U$ be the closed subscheme whose ideal sheaf is $\operatorname{\mathfrak{m}}^{2}\subseteq\operatorname{\mathcal{O}}_{U}$. Then for any $d\in\operatorname{\mathbb{Z}}_{\geq 0}$ we have

[TABLE]

Proof.

This is Lemma 2.2 of [Poo08]; the condition on the dimension is not needed here. ∎

Lemma \thelemma (Singularities of low degree).

For $\mathcal{P}_{r}$ defined as in section 2,

[TABLE]

Proof.

The proof is parallel to the one of Lemma 2.4 in [Wut16], just use section 2.1 above and $(U-V)_{<r}=U_{<r}-V$ instead of $U_{<r}$. ∎

Remark \therem.

Note that the condition on the dimension of $V$ is not needed here, so this proof would work also for points in $V$, and we would get the same density as in Lemma 2.3 of [Poo08]:

[TABLE]

Nevertheless, we cannot ask for smoothness in points on $V$ for the polynomials in $\mathcal{P}_{r}$ or $\mathcal{P}$, since the assumption on the dimension in our case, i.e. $2l\leq m+k$, does not imply convergence of the zeta function.

2.2. Points of medium degree

Lemma \thelemma (Singularities of medium degree).

For $\mathcal{Q}_{r}^{\operatorname{medium}}$ defined as in section 2, we have $\lim\limits_{r\rightarrow\infty}\mu_{Z}(\mathcal{Q}_{r}^{\operatorname{medium}})=0$.

Proof.

This is parallel to the proof of Lemma 3.2. of [Poo08], just ignore points in $V$. Again, the condition on the dimension is not used to prove this, but it does not work for points in $V$ as the resulting corresponding series does not converge for $2l>m$. ∎

2.3. Points of high degree

Lemma \thelemma (Singularities of high degree off V).

For $\mathcal{Q}^{\operatorname{high}}_{U-V}$ defined as in section 2,

[TABLE]

Proof.

This is Lemma 4.2. of [Poo08]. The assumption $m>2l$ is not used in the proof. ∎

Lemma \thelemma (Singularities of high degree on V).

For $\mathcal{Q}_{V}$ defined as in section 2,

[TABLE]

Proof.

The proof of this claim uses the induction argument used by Poonen in the proof of Lemma 2.6 in [Poo04], and up to the definition of the polynomials $g_{i}$, it is similar to the one of Lemma 4.3 of [Poo08].

We may assume $U$ is contained in $\operatorname{\mathbb{A}}^{n}=\left\{x_{0}\neq 0\right\}\subseteq\operatorname{\mathbb{P}}^{n}$. Dehomogenize by setting $x_{0}=1$, and identify $S_{d}$ with the space of polynomials $S_{d}^{\prime}\subseteq\operatorname{\mathbb{F}}_{q}\left[x_{1},\ldots,x_{n}\right]=A$ of degree $\leq d$ and $I_{d}$ with a subspace $I_{d}^{\prime}\subseteq S_{d}^{\prime}$.

Let $P$ be a closed point of $U$. Choose a system of local parameters $t_{1},\ldots,t_{n}\in A$ at $P$ on $\operatorname{\mathbb{A}}^{n}$ such that $t_{m+1}=\ldots=t_{n}=0$ defines $U$ locally at $P$, and $t_{1}=\ldots=t_{m-l}=t_{m+1}=\ldots=t_{n}=0$ defines $V$ locally at $P$. We may assume in addition that $t_{1},\ldots,t_{m-l}$ vanish on $Z$ (cf. [Poo08], Lemma 4.3). By definition, $dt_{1},\ldots,dt_{n}$ are a $\operatorname{\mathcal{O}}_{\operatorname{\mathbb{A}}^{n},P}$-basis for the stalk $\Omega^{1}_{\operatorname{\mathbb{A}}^{n}|\operatorname{\mathbb{F}}_{q},P}$. Let $\partial_{1},\ldots,\partial_{n}$ be the dual basis of the stalk of the tangent sheaf $\mathcal{T}_{\operatorname{\mathbb{A}}^{n}|\operatorname{\mathbb{F}}_{q}}$ at $P$. Choose $s\in A$ satisfying $s(P)\neq 0$ to clear denominators such that $D_{i}=s\partial_{i}$ defines a global derivation $A\rightarrow A$ for all $i$. Then there exists a neighbourhood $N_{P}$ of $P$ in $\operatorname{\mathbb{A}}^{n}$ such that $N_{P}\cap\left\{t_{m+1}=\ldots=t_{n}=0\right\}=N_{P}\cap U$, $\Omega^{1}_{N_{P}|\operatorname{\mathbb{F}}_{q}}=\oplus_{i=1}^{n}\operatorname{\mathcal{O}}_{N_{P}}dt_{i}$ and $s\in\operatorname{\mathcal{O}}(N_{P})^{*}$. We can cover $U$ with finitely many $N_{P}$, so we may assume that $U$ is contained in $N_{P}$ for some $P$. For $f\in I_{d}^{\prime}\cong I_{d}$, the hypersurface section $H_{f}\cap U$ fails to be smooth of dimension $m-1$ at some point $P\in V$ if and only if $(D_{1}f)(P)=\ldots=(D_{m}f)(P)=0$.

Let $\tau=\max_{i}(\deg t_{i})$, $\gamma=\left\lfloor(d-\tau)/p\right\rfloor$, and $\nu=\left\lfloor d/p\right\rfloor$. If we choose $f_{0}\in I^{\prime}_{d}$, $g_{1}\in S_{\gamma}^{\prime},\ldots,g_{l-k}\in S_{\gamma}^{\prime}$ uniformly at random, then

[TABLE]

is a random element of $I_{d}^{\prime}$, because of $f_{0}$ and since by assumption $l-k\leq m-l$. Note that we cannot define more polynomials $g_{l-k+1},\ldots,g_{m}$ as in [Poo08], since our condition on the dimension would not yield a polynomial in $I_{d}^{\prime}$.

For $i=0,\ldots,l-k$ we define

[TABLE]

By definition of the $D_{i}$, this subscheme depends only on $f_{0},g_{1},\ldots,g_{i}$.

As in Lemma 2.6 of [Poo04], one can show that for $0\leq i\leq l-k-1$, conditioned on a choice of $f_{0},g_{1},\ldots,g_{i}$ for which $\dim(W_{i})\leq l-i$, the probability that $\dim(W_{i+1})\leq l-i-1$ is $1-o(1)$ as $d\rightarrow\infty$.

It follows that for $i=0,\ldots,l-k$ we have $\operatorname{\mbox{Prob}}(\dim W_{i}\leq l-i)=1-o(1)$ as $d\rightarrow\infty$ and thus $W_{l-k}$ is at most of dimension $k$ with probability $1-o(1)$, which is what we claimed, since $W_{l-k}$ contains the points where $H_{f}\cap U$ is not smooth. ∎

Proof of 2.1.

We have the inclusions

[TABLE]

The first inclusion is clear. For the second, let $f\in\mathcal{P}_{r}$. If $f$ is not in $\mathcal{P}$, then by definition, either $H_{f}\cap U$ is not smooth at some point $P\in U-V$, or $\dim(H_{f}\cap U)_{\operatorname{sing}}\geq k+1$. In the first case, this point $P$ must be of some degree $\geq r$, since $f\in\mathcal{P}_{r}$, and $f\in\mathcal{Q}_{r}^{\operatorname{medium}}\cup\mathcal{Q}_{U-V}^{\operatorname{high}}$. For the second case, if $H_{f}\cap U$ is smooth at all points in $U-V$, then the singular locus of the hypersurface section is completely contained in $V$ and $f\in\mathcal{Q}_{V}$. If $H_{f}\cap U$ is not smooth at some point in $U-V$, then we are in the first case again.

By section 2.1, 2.2, 2.3 and 2.3,

[TABLE]

∎

3. Results for V not smooth

In the condition on the dimension we will need for the analogue of 2.1 in the case where $V$ is not smooth, the embedding dimension of a scheme $X$ at a point $P$ will occur naturally. It is defined as $e(P)=\dim_{\kappa(P)}(\Omega^{1}_{X|\operatorname{\mathbb{F}}_{q}}(P))$. Let

[TABLE]

be the subscheme such that a scheme morphism $f:T\rightarrow X$ factors through $X_{e}$ if and only if $f^{*}\Omega^{1}_{X|\operatorname{\mathbb{F}}_{q}}$ is locally free of rank $e$. Then $X_{e}$ is the locally closed subscheme of $X$ where the embedding dimension of $X$ is $e$.

The condition on the dimension for $V$ non smooth in the Bertini smoothness theorem (cf. [Wut16], [Gun15]) for hypersurface sections containing a closed subscheme is $\max\{e+\dim V_{e}\}<m$ instead of $2\dim<\dim X$ for the case $V$ smooth ([Poo08]). Hence, one would expect an analogue of 2.1 in the case $V$ non smooth to hold for $\max\{e+\dim V_{e}\}\leq m+k$. But using our methods, we cannot prove this exact analogue of 2.1 where $V$ is not smooth, since we can only bound the dimension of the bad points of high degree in each $V_{e}$ and not in $V$. The only case that still works is $k=0$:

Theorem 3.1**.**

Let $X$ be a quasi-projective subscheme of $\operatorname{\mathbb{P}}^{n}$ and $Y$ a finite subscheme of $\operatorname{\mathbb{P}}^{n}$ such that $U=X-(X\cap Y)$ is smooth of dimension $m\geq 0$ over $\operatorname{\mathbb{F}}_{q}$. Let $Z$ be a closed subscheme of $\operatorname{\mathbb{P}}^{n}$ such that $Z\cap Y=\emptyset$. Let $T\subseteq H^{0}(Y,\mathcal{I}_{Z}\cdot\operatorname{\mathcal{O}}_{Y})$. Assume that for $V=Z\cap U$ we have $\max\{e+\dim V_{e}\}\leq m$. Define

[TABLE]

Then $\mu_{Z}(\mathcal{P})=\dfrac{\#T}{\#H^{0}(Y,\operatorname{\mathcal{O}}_{Y})}\zeta_{U-V}(m+1)^{-1}$.

The proof again uses the closed point sieve. We define $\mathcal{P}_{r}$, $\mathcal{Q}_{r}^{\operatorname{medium}}$ and $\mathcal{Q}_{U-V}^{\operatorname{high}}$ as in in the previous section. The proofs for 2.1, 2.2 and 2.3 do not need the conditions on the smoothness or dimension of $V$, and thus we only have to show the following lemma on singularities of high degree:

Lemma \thelemma (Singularities of high degree for $V$ not smooth).

Define

[TABLE]

Then $\mu_{Z}(\mathcal{Q})=0$.

Proof.

We may assume $U$ is contained in $\operatorname{\mathbb{A}}^{n}=\left\{x_{0}\neq 0\right\}\subseteq\operatorname{\mathbb{P}}^{n}$. As there are only finitely many $V_{e}$, it is enough to bound the probability that a polynomial gives a singular locus of dimension $\geq 1$ if we intersect this singular locus with $V_{e}$.

Let $P$ be a closed point of $V_{e}$. Since $U$ is smooth, we can choose a system of local parameters $t_{1},\ldots,t_{n}\in A$ on $\mathbb{A}^{n}$ such that $t_{m+1}=\ldots=t_{n}=0$ defines $U$ locally at $P$. Then $dt_{1},\ldots,dt_{n}$ are a basis for the stalk of $\Omega^{1}_{\operatorname{\mathbb{A}}^{n}|\operatorname{\mathbb{F}}_{q}}$ at $P$ and $dt_{1},\ldots,dt_{m}$ are a basis for the stalk of $\Omega^{1}_{U|\operatorname{\mathbb{F}}_{q}}$ at $P$. Using the exact sequence ([Har93], Section II.8) $\mathcal{I}_{V}/\mathcal{I}_{V}^{2}\rightarrow\Omega_{X}^{1}\otimes\mathcal{O}_{V}\stackrel{{\scriptstyle\phi}}{{\rightarrow}}\Omega_{V}^{1}\rightarrow 0,$ we show that $dt_{1},\ldots,dt_{m-e}$ form a basis of the kernel of $\phi$ at $P$ and $dt_{m-e+1},\ldots,dt_{m}$ a basis of $\Omega^{1}_{V|\operatorname{\mathbb{F}}_{q},P}\otimes\operatorname{\mathcal{O}}_{V_{e},P}$. In particular, $t_{1},\ldots,t_{m-e}$ all vanish on V, since $\Omega^{1}_{V|\operatorname{\mathbb{F}}_{q}}\otimes\kappa(P)\cong\operatorname{\mathfrak{m}}_{V,P}/\operatorname{\mathfrak{m}}_{V,P}^{2}$. Again, we may also assume in addition that $t_{1},\ldots,t_{m-e}$ vanish on $Z$.

As in section 2.3, we define $\partial_{i}$, $s$ and $D_{i}$ such that for $f\in I_{d}^{\prime}\cong I_{d}$, the hypersurface section $H_{f}\cap U$ fails to be smooth of dimension $m-1$ at some point $P\in V_{e}$ if and only if $(D_{1}f)(P)=\ldots=(D_{m}f)(P)=0$.

Let $\tau=\max\limits_{1\leq i\leq{l_{e}}-1}(\deg t_{i})$ and $\gamma=\lfloor(d-\tau)/p\rfloor$ where $l_{e}=\dim V_{e}$. We select $f_{0}\in I^{\prime}_{d}$ and $g_{1}\in S^{\prime}_{\gamma},\ldots,g_{l_{e}-1}\in S^{\prime}_{\gamma}$ uniformly and independently at random. Then the distribution of

[TABLE]

is uniform over $I^{\prime}_{d}$, since by assumption, $l_{e}\leq m-e$.

Now everything works as in the proof of section 2.3: for $i=0,\ldots,l_{e}$ we define

[TABLE]

depending only on $f_{0},g_{1},\ldots,g_{i}$. Using the induction as in Lemma 2.6 of [Poo04], we show that $W_{l_{e}}$ is at most of dimension 0 with probability $1-o(1)$, which is what we claimed. ∎

Proof of 3.1.

As in section 2, we have the inclusions

[TABLE]

By 2.2, 2.3 and 3, the error in our approximation is negligible, and section 2.1 yields 3.1. ∎