3
$\begingroup$

Let $X$ be a normal projective variety over an algebraically closed field $k$. Given any morphism $f:Y\to X$, there is a pullback homomorphism $f^*:\text{Cl}(X)\to\text{Cl}(Y)$, where $\text{Cl}(X)$ denotes the (Weil) divisor class group of $X$ (see for example Fulton's "Intersection Theory" section 6.2 for a definition of $f^*$, where it is called a Gysin homomorphism).

Question. Let $D\in\text{Cl}(X)$ be nontrivial. Does there exist a smooth projective curve $C$ and a morphism $i:C\to X$ such that $i^*(D)\in \text{Cl}(C)$ is nontrivial?

Ideally, I would like an answer that can be used to explicitly construct $i$ and $C$ given $X$ and $D$. But I will accept any answer to the existence question, as long as it handles the case that $X$ is not smooth and $k$ has positive characteristic. Note that my question is slightly weaker than asking for $i^*$ to be injective (I just want one element $D$ to not be in the kernel), though a construction of $i$ and $C$ that does not depend on $D$ would be great!

Partial answer.

The answer is yes when $X$ is smooth. In this case the divisor class group is isomorphic to the Picard group, and a solution is outlined by this Mathoverflow question and answer. Namely, successively replace $X$ with any smooth hyperplane section. This induces an injection of Picard groups by Grothendieck-Lefschetz, unless $\text{dim}(X)=2$. At this point one may instead need to use an element of the linear series of a sufficiently large multiple of a hyperplane.

If $X$ is just normal but not smooth, the answer is still yes if we have a resolution of singularities $r:\widetilde{X}\to X$ (this always holds in characteristic zero, and conjecturally holds for all characteristics). If $Y\subseteq X$ is the singular locus, then the map $$\text{Cl}(X)\xrightarrow{r^*}\text{Cl}(\widetilde{X})\to\text{Cl}(\widetilde{X}-r^{-1}(Y))$$ also factors as $$\text{Cl}(X)\xrightarrow{\cong}\text{Cl}(X-Y)\xrightarrow{\cong}\text{Cl}(\widetilde{X}-r^{-1}(Y)),$$ with the first map being an isomorphism because $Y$ has codimension at least $2$ in $X$. In particular, $r^*$ is injective, so $r^*(D)$ is nontrivial. Applying the result for the smooth variety $\widetilde{X}$, there is some $i:C\to\widetilde{X}$ such that $(r\circ i)^*(D)$ is nontrivial, solving the question for $X$ as well. (This is a special case of the argument used by Ravindra and Srinivas section 1 of their paper proving a version of Grothendieck-Lefschetz for normal varieties in characteristic 0.)

I'm interested in the positive characteristic setting, where resolution of singularities is still conjectural so the above argument doesn't apply. We could try using an alteration $j:X'\to X$ instead, but unfortunately, the map $j^*$ on divisor classes is no longer guaranteed to be injective in this case; if $D$ happens to be in the kernel then we're stuck.

Speaking a bit less formally, it feels like resolution of singularities is a lot stronger than we really need for this problem. A typical $i:C\to X$ isn't going to come anywhere near most points in the singular locus of $X$, so it seems unreasonable that we would have to repair the entire variety first if we only end up using such a small portion of it. On the other hand, I do know that singularities in positive characteristic can be particularly nasty, so maybe there's not much that can be done without first moving over to a smooth variety. Are there relevant things we can say about homomorphisms between divisor class group of normal varieties that don't require resolution of singularities?

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.