Conormal Variety and fiber of a projection

Question

Let be $X$ a projective irreducible variety in $\mathbb{P}^{N}$ of dimension $n$. Let be $ V_0 = \{(p,H)\in\mathbb{P}^N\times (\mathbb{P}^{N})^{*} \mid p\in X_{sm},\ T_p X \subset H\}$ and $V$ its Zariski closure. Consider the projections $\pi_{1}:V\rightarrow X$ and $\pi_{1,0}:V_0\rightarrow X_{sm}$. I know that for $p\in X_{sm}$ we have $(\pi_{1,0})^{-1}(p)=\{p\} \times \Lambda(T_p X)$, where $\Lambda(T_p X)$ is the linear system of hyperplanes containing $T_p X$. What about the fiber of $\pi_1$? In particular i have to prove that if $p \in X_{sm}$ then the fiber is an irreducible variety of dimension $N-n-1$.

Answer 1

$\newcommand{\PP}{{\mathbb P}}$ $\newcommand{\Xsm}{{X_{\mathrm{sm}}}}$

The fiber $\pi_1^{-1}(p)$ is equal to $\pi_{1,0}^{-1}(p)$ for $p \in \Xsm$. Consider the map $q:Y = X \times (\PP^N)^* \to X$. Let $U$ be an open subset in the open set $\Xsm \subseteq X$ with $p \in U$. Then $\pi_{1,0}^{-1}(U)$ is a closed subset of the open set $q^{-1}(U)$ (it is given by a certain set of equations, see below), so it is an intersection $A \cap q^{-1}(U) = V_0 \cap q^{-1}(U)$ with a closed set $A$ in $Y$. Now $A' = Y - q^{-1}(U)$ is a second closed set in $Y$. It is $V_0 \subseteq A \cup A'$ and therefore the closure $V \subseteq A \cup A'$. But $(A \cup A') \cap q^{-1}(U) = V_0$ and therefore $q^{-1}(U) \cap V = q^{-1}(U) \cap V_0$. But this gives the contended $\pi^{-1}(p) = \pi_{1,0}^{-1}(p)$.

Now let $X = V(f_1,\ldots,f_s)$ with $f_i \in k[x_0,\ldots,x_N]$, homogeneous, generating the ideal of $X$. Then

$$V_0 \cap q^{-1}(U) = \{ (p,H) \in (U,(\PP^N)^*) \mid H = V(\sum_{i=0}^N c_i x_i) \text{ and } \mathrm{rank} M = N - n \}$$

where $M$ is a matrix with $s +1$ rows and $N+1$ columns, having $(\frac{\partial f_i}{\partial x_j})_{ij}$ as the first $s$ rows and $N+1$ columns and $(c_0 \cdots c_N)$ as the row $s+1$ and $N+1$ columns. For $p \in \Xsm$ the matrix $(\frac{\partial f_i}{\partial x_j})|_p$ has rank $N - n$ by the jacobian criterion of regularity. Therefore our rank condition says, that $H=V(\sum c_i x_i)$ contains the tangential plane $T_p X$, which is the null-space of $(\frac{\partial f_i}{\partial x_j})|_p$. So we have proved, that $V_0 \cap q^{-1}(U) = A \cap q^{-1}(U)$ for a closed set $A$.

Now to your last question, as $\pi_1^{-1}(p) = \pi_{1,0}^{-1}(p)$ for $p \in \Xsm$ as we have proved above, we have $n + 1$ conditions for the hyperplanes $H$ in the fiber: they must go through $p$ (one condition) and they must contain $n$ linear independent vectors from $T_p X$ ($n$ conditions). These conditions cut out a linear space in the space of hyperplanes, which is of dimension $N - (n +1)$ and, as a linear subspace, irreducible.