# Lefschetz type theorems for linear sections

Let $$X\subset\mathbb{P}^n$$ be e normal variety, $$L\subset\mathbb{P}^n$$ a linear subspace, and $$Y = X\cap L$$ a linear section. Assume that $$Y$$ is also normal. In particular, we have that $$Sing(X)$$ has codimension at least two in $$X$$, and $$Sing(Y)$$ has codimension at least two in $$Y$$.

Does there exist any generalization of the Lefschetz hyperplane theorem to higher codimension linear sections of singular varieties ensuring that the restriction map

$$Cl(X)\rightarrow Cl(Y)$$

is an isomorphism or at least surjective?

• You would need some assumptions. For instance, a rank 5 quadric $X$ in $\mathbb{P}^n$ has $\operatorname{Cl}(X)= \mathbb{Z}$, but it admits a hyperplane section $Y$ of rank 4, hence $\operatorname{Cl}(Y)=\mathbb{Z}^2$. – abx May 16 at 8:02
• In your example we have $Sing(X) = Sing(Y)$ if I got it right. In the case I am interested in I know that $Sing(X) = Sing(Y)\cap L$ but $Sing(X)\neq Sing(Y)$. – Jessica_90 May 16 at 8:16
• Look for Grothendieck-Lefschetz theorey, for instance this paper: arxiv.org/pdf/1601.05846.pdf – Hailong Dao May 16 at 16:28
• In Theorem 1 (v) here dima.unige.it/~badescu/attivita%20scientifica/… they say that if $Y\subset \mathbb{P}^n$ is normal of dimension at least three and $Y$ can be set-theoretically defined by at most $n-3$ equations then $Pic(\mathbb{P}^n)\rightarrow Pic(Y)$ is an isomorphism. Is not this in contradiction with abx example? We could take for instace a quadric cone with vertex a point in $\mathbb{P}^4$. – Jessica_90 May 17 at 6:27
• No contradiction. $\operatorname{Pic}(Y)$ is not the same thing as $\operatorname{Cl}(Y)$. – abx May 17 at 14:28