I am looking for a reference to the following fact: monodromy group acting on the cohomology of smooth hyperplane sections of a smooth projective variety $X$ over $\mathbb C$ is preserved under isomorphic projection.


To be precise, suppose that $X\subset \mathbb P^N$ is a smooth projective variety of dimension $n+1$ and $Y$ is its smooth hyperplane section. As $Y$ varies in the class of smooth hyperplane sections of $X$, a monodromy action on $H^n(Y,\mathbb Z)$ arises; by monodromy group of $X$ I will mean the corresponding subgroup of $\mathrm{Aut}(H^n(Y,\mathbb Z))$.


Suppose now that there exists a pont $p\in\mathbb P^N\setminus Y$ such that the projection $\pi\_p\colon X\to X'\subset\mathbb P^{N-1}$ is an isomorphism. Do you know a reference to the effect that this monodromy group of $X'$ is the same as that of $X$?


Thank you in advance,  

Serge



Reference request: monodromy and isomorphic projections
The comments above are too long, so I am replacing them by an answer. The reference for most such results in algebraic geometry is SGA 7\_2. I am sure that the following is in there somewhere.


Your assertion follows from a more general statement. Let $X\subset \mathbb{P}^N$ be a smooth projective variety. Let $\widehat{\mathbb{P}}^N$ denote the dual projective space parameterizing hyperplanes in $\mathbb{P}^N$. Let $\widehat{X}\subset \mathbb{P}^N$ be the dual variety of $X$, i.e., the image of the natural map to $\widehat{\mathbb{P}}^N$ from the projectivized normal bundle of $X$ in $\mathbb{P}^N$, $\mathbb{P}\_X \mathcal{N}\_{X/\mathbb{P}^N}$. Since the projectivized normal bundle has dimension $N-1$, the image $\widehat{X}$ is a proper subvariety; typically it is a hypersurface. Denote by $B\subset \widehat{X}$ the closed locus over which the morphism $\mathbb{P}\_X\mathcal{N}\_{X/\mathbb{P}^N} \to \widehat{X}$ is not an isomorphism. In characteristic $0$, always $B$ has codimension $\geq 2$ in $\widehat{\mathbb{P}}^N$. In characteristic $p$, there are sufficient conditions to assure this, but usually people just re-embed $X$ in a bigger projective space by the $d$-uple Veronese map for $d\geq 2$; then $B$ will have codimension $\geq 2$.


A **Lefschetz pencil** is a line $\ell \in \widehat{\mathbb{P}}^N$ that is disjoint from the codimension $2$ subset $B$ and that intersects $\widehat{X}$ (tangentially) transversally at every intersection point. When $B$ has codimension $2$, Lefschetz pencils exist (by the same sort of parameter count that establishes Bertini's smoothness theorem). All of the above is in SGA 7\_2.


The assertion is that for every Lefschetz pencil $\ell$, the induced homomorphism $$f\_{X,\ell}:\pi\_1(\ell\setminus \ell\cap \widehat{X}) \to \pi\_1(\widehat{\mathbb{P}}^N\setminus \widehat{X})$$ is surjective. This proves your assertion as follows. For a point $p$ such that the linear projection from $p$ restricts to an isomorphism from $X$ to its image $Y$ in $\mathbb{P}^{N-1}$, the corresponding hyperplane $H\_p\subset \widehat{\mathbb{P}}^N$ equals $\widehat{\mathbb{P}}^{N-1}$ and $H\_p\cap \widehat{X}$ equals $\widehat{Y}$. In characteristic $0$ at least, existence of a Lefschetz pencil implies that there exists $\ell\in H\_p$ that is a Lefschetz pencil. Thus, we have a composition of group homomorphisms, $$\pi\_1(\ell \setminus \ell \cap \widehat{X}) \xrightarrow{f\_{Y,\ell}} \pi\_1(H\_p\setminus H\_p\cap \widehat{X}) \xrightarrow{f\_{X,Y}} \pi\_1(\widehat{\mathbb{P}}^N\setminus \widehat{X}).$$ Since the composition $f\_{X,Y}\circ f\_{Y,\ell}$ is surjective, also $f\_{X,Y}$ is surjective. Thus, for every group homomorphism $\phi:\pi\_1(\widehat{\mathbb{P}}^N\setminus \widehat{X}) \to G$, the image of $\phi$ equals the image of $\phi\circ f\_{X,Y}$. 


Thus it suffices to prove that for every Lefschetz pencil $\ell$, $f\_{X,\ell}$ is surjective. Fix a point $p$ of $\ell$. Denote by $U\subset \widehat{\mathbb{P}}^N\setminus (\widehat{X}\cup \{p\})$ the set of points $q$ such that line $\ell\_q$ spanned by $p$ and $q$ is a Lefschetz pencil. The transversality conditions in the definition of a Lefschetz pencil are open conditions. Thus, $U$ is an open subset. Since for every $q\in \ell\setminus\{p\}$, $q$ is in $U$, $U$ is a nonempty open subset. Thus, $U$ is a dense open subset of $\widehat{\mathbb{P}}^N\setminus (\widehat{X}\cup \{p\})$. In particular, $U$ is connected. Thus, the images of the homomorphisms $f\_{X,\ell\_q}$ for $q\in U$ are all equal to a common subgroup $\Gamma$ of $\pi\_1(\widehat{\mathbb{P}}^N\setminus \widehat{X},p)$. 


Let $\rho:Y\to \widehat{\mathbb{P}}^N\setminus \widehat{X}$ be the covering space associated to the standard left action of $\pi\_1(\widehat{\mathbb{P}}^N\setminus \widehat{X},p)$ on the set of right $\Gamma$-cosets. The restriction of this left action to the subgroup $\Gamma$ has a fixed point: the neutral coset $\Gamma$. Thus, the restriction of $\rho$ over every line $\ell\_q\setminus \ell\_q\cap \widehat{X}$ has a cross section. The union of all of these cross sections defines a cross section of the restriction of $\rho$ over the open dense subset $U$. For every point $r\in \widehat{\mathbb{P}}^N\setminus \widehat{X}$, there exists a neighborhood $B\_\epsilon(r)$ of $r$ in this open over which $\rho$ is trivialized. Since the complement of $U$ is an analytic subvariety of complex codimension $1$, real codimension $2$, $B\_\epsilon(r)\cap U$ is nonempty and connected. Thus, the cross section of $\rho$ over $U$ extends uniquely to a cross section over all of $B\_\epsilon(r)$. Since this holds for every $r$, the cross section extends over all of $\widehat{\mathbb{P}}^N\setminus \widehat{X}$. It follows that the neutral coset $\Gamma$ is a fixed point for all of $\pi\_1(\widehat{\mathbb{P}}^N\setminus \widehat{X},p)$. Thus, $\Gamma$ is the entire group.


