While reading Fulton's Intersection theory, I came across the following comment.

Let $X$ be a projective scheme over an algebraically closed field. Assume we have been given a map $g : \mathbb{P}^1 \rightarrow S^nX$. Then this map factors though $S^nC$, where $C$ is a smooth curve with a proper map $C \rightarrow X$ (Example 1.6.3).

What I can achieve is a curve $C' \xrightarrow{i} X$ such that $g$ factors through $S^nC$, i.e. there exists $g' : \mathbb{P}^1 \rightarrow S^nC'$ such that $i \circ g' = g$. Of course, I have a normalization $\tilde{C'} \xrightarrow{\pi} C'$. The problem is, it may not be possible to lift $g'$ to $S^n\tilde{C'}$.

Thanks in advance.

  • $\begingroup$ @abx Don't you need that it is dominant? $\endgroup$ – random123 Nov 30 '17 at 15:19
  • $\begingroup$ Yes, you are right. I deleted my comment. $\endgroup$ – abx Nov 30 '17 at 16:14

Let me try the following argument.

First, as you explain one can reduce the general case to the case when $X$ is a curve $C$, possibly singular and reducible. This is because the preimage of $g(\mathbb{P}^1)$ under the quotient morphism $\phi: X^n \to S^n(X)$ has dimension dimension one (or zero if $g$ is constant), and one can find a closed reduced curve $C \subset X$ such that the image $g(\mathbb{P}^1)$ is contained in $S^n(C) \subset S^n(X)$, and thus $g$ will factor through $S^n(C)$.

We need to show that $g: \mathbb{P}^1 \to S^n(C)$ factors through $S^n(\tilde{C}) \to S^n(C)$ where $\tilde{C} \to C$ is the normalization. We can reduce to the case when $g: \mathbb{P}^1 \to S^n(C)$ represents divisors on $C$ with no fixed components. Indeed, if $x \in C$ is a point in the support of every divisor in the family, the morphism $g$ will factor as a composition $\mathbb{P}^1 \to S^{n-1}(C) \to S^{n}(C)$, where the latter map maps an effective divisor $D$ on $C$ to $D + x$. This way we can get rid of all fixed components while decreasing $n$.

Finally, if $g$ has no fixed components, then for every point $x \in C$ the set of $t \in \mathbb{P}^1$ such that $x$ is contained in the support of $g(t)$ is finite. In particular, if we let $U \subset \mathbb{P}^1$ to denote the subset parametrizing divisors supported at the smooth locus of $C$, the set $U$ will be open and non-empty. Furthermore it is clear that we have a morphism $U \to S^n(\tilde{C})$ lifting $g|_U$, because normalization is as isomoprhism away from the singular locus of $C$. By the valuative criterion of properness, we can extend $U \to S^n(\tilde{C})$ to a morphism $\tilde{g}: \mathbb{P}^1 \to S^n(\tilde{C})$, and it will automatically be a lifting of $g$ (since it is such on an open dense subset).


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