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.

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    $\begingroup$ 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. $\endgroup$ – Jason Starr Dec 13 '19 at 15:44
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    $\begingroup$ @JasonStarr: Thanks! That's fantastic. If you posted it as an answer, I would be delighted to accept it. $\endgroup$ – Tina Dec 14 '19 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).

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    $\begingroup$ 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. $\endgroup$ – Tina Dec 12 '19 at 4:25

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