# Multisections of the universal curve

Fix some $$g \geq 2$$. Let $$\mathcal{M}_g$$ be the moduli space of smooth genus $$g$$ curves over $$\mathbb{C}$$. For some $$d \geq 1$$, let $$X_{g,d} \rightarrow \mathcal{M}_g$$ be the family whose fiber over $$S \in \mathcal{M}_g$$ is the $$d^{\text{th}}$$ symmetric power of $$S$$. I'm aware that due to the presence of curves with automorphisms this doesn't (strictly speaking) exist, but let's ignore that point (e.g. by adding a full level structure to $$\mathcal{M}_g$$ to rigidify things).

Question 1: Does there exist some $$d \geq 1$$ such that $$X_{g,d} \rightarrow \mathcal{M}_g$$ has a section? I expect something like Weierstrass points will work here, but I don't know how they vary in families.

Question 2: Assuming that Question 1 has a positive answer, what I'm really interested in is the following. Does there exit some $$d,e \geq 1$$ such that there exist sections $$\sigma\colon \mathcal{M}_g \rightarrow X_{g,d}$$ and $$\sigma'\colon \mathcal{M}_g \rightarrow X_{g,e}$$ such that for all $$S \in \mathcal{M}_g$$, the $$d$$ points making up $$\sigma(S)$$ are disjoint from the $$e$$ points making up $$\sigma'(S)$$? Here I don't have a candidate for the two disjoint multisections.

• Welcome new contributor. Question 2 has a negative answer. The Franchetta Conjecture, proved by John Harer, says that for every section $\sigma$, the corresponding divisor on the universal curve over $\mathcal{M}_g$ has divisor class equal to a positive integer multiple of the relative canonical class and the pullback of a multiple of the lambda class. Now compute the intersection of two such divisors, for instance, on the surface that is the total space over a "Satake curve", i.e., a general complete intersection curve for the Satake compactification. Dec 13, 2019 at 15:44
• @JasonStarr: Thanks! That's fantastic. If you posted it as an answer, I would be delighted to accept it.
– Tina
Dec 14, 2019 at 15:37

I would like to give an answer to the simple Question 1. The answer relies on the following lemma.

Lemma. For any fixed $$(r,g,n)$$ there exists a hypersurface $$V\subset \mathbb CP^n$$ that doesn't contain any smooth genus $$g$$ curve of degree $$\le r$$.

I think, this lemma can be proven by simple dimension count.

Now, to solve the question, find $$n$$, such that (the universal curve) $$M_{g,1}$$ can be embedded in $$\mathbb CP^n$$. Let $$r$$ be the degree of all the genus $$g$$ curves in the embedding. Take, a hypersurface $$V$$ from the Lemma and let $$d$$ be its degree. Now, take and the intersection of $$V$$ with each curve. This will give you the desired section.

(As for Question 2, I have no clue).

• Thanks! This is a nice way to deal with Question 1 (I was too focused on trying to do it by picking special points on a curve). Since my main interest is in the followup Question 2, I'm going to hold off on accepting it.
– Tina
Dec 12, 2019 at 4:25