# Reference request: monodromy and isomorphic projections

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 $X$?

Serge

• This follows from the assertion that for the dual projective space $\widehat{\mathbb{P}}^N$, for the dual variety $\widehat{X}\subset \widehat{\mathbb{P}}^N$ (typically a hypersurface unless $X$ is a linear subspace ...), for the singular locus $B\subset \widehat{X}$, for every line $\ell \subset \widehat{\mathbb{P}^N}$ that is disjoint from $B$ and that intersects $\widehat{X}$ transversally, the homomorphism $\pi_1(\ell\setminus \ell\cap \widehat{X}) \to \pi_1(\widehat{\mathbb{P}}^N\setminus \widehat{X})$ is surjective. This is straightforward to prove . . . Mar 29, 2017 at 9:33
• . . . The fundamental group of $\pi_1(\ell\setminus \ell\cap \widehat{X})$ is a free group on the generators arising from intersection points $\widehat{X}$ with a single relation equal to the product of these generators (with appropriate orientation). Thus a homomorphism from this fundamental group to a group $G$ is trivial if and only if every generator maps to the identity. For a homomorphism from the fundamental group of $\widehat{\mathbb{P}}^N\setminus \widehat{X}$ to $G$, the restriction as above is trivial if and only if there is trivial "ramification" along $\widehat{X}$ . . . Mar 29, 2017 at 9:38
• . . . In that case, by Seifert-van Kampen, the homomorphism is the restriction of a homomorphism from the fundamental group $\pi_1(\widehat{\mathbb{P}}^N\setminus B)$ to $G$. Since $B$ has complex codimension $2$, thus real codimension $4$, this equals $\pi(\widehat{\mathbb{P}}^N)$, which is trivial. Hmm, now I realize that this only proves that the image of $\pi_1(\ell \setminus \ell\cap \widehat{X})$ is contained in no proper normal subgroup of $\pi_1(\widehat{\mathbb{P}}^N\setminus \widehat{X})$ . . . Mar 29, 2017 at 9:44
• The argument above can be corrected. Fix a general point $p$ of $\widehat{\mathbb{P}}^N$. Let $U\subset \widehat{\mathbb{P}}^N\setminus (\widehat{X}\cap\{p\})$ be the open subset of all points $q$ such that line $\ell$ spanned by $p$ and $q$ satisfies the transversality conditions. This is a connected open subset whose complement is a proper complex analytic subvariety of complex codimension $1$, real codimension $2$. As $q$ varies, the images in $\pi_1(\widehat{\mathbb{P}}^n\setminus\widehat{X})$ are all equal, by continuity and connectedness. . . . Mar 29, 2017 at 9:52
• . . . Consider the covering space of $\widehat{\mathbb{P}}^n\setminus \widehat{X}$ obtained from the coset space of this common subgroup $\pi_1(\ell \setminus \ell\cap \widehat{X},p)$ inside $\pi_1(\widehat{\mathbb{P}}^N\setminus \widehat{X},p)$. Restricted over every line $\ell$, this covering has a unique cross section coming from the neutral coset. As $q$ varies, the union of these cross sections defines a cross section of the covering space when restricted over $U$. Now use that the complement of $U$ is an analytic subvariety of real codimension $2$ to see that the cross section extends. Mar 29, 2017 at 9:56

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.

• Dear Jason, thank you for your time. Most certainly this assertion is absent from SGA7_2, otherwise I would not have asked for a reference:) Anyway, I guess one may safely assume that the required reference does not exist. Thank you again. Mar 29, 2017 at 16:05