Let $C$ be a smooth curve of genus $3$ over $\mathbb{C}$. Is it true that there exist $P\neq Q \in C$ such that $4P \sim 4Q$ ? ($\sim$ denotes linear equivalence)
Notice that if $C$ is hyperelliptic then this is true (just take two different points fixed by the hyperelliptic involution). My (very optimistic) guess is that the converse should hold. If we denote by $f \in K(C)$ the function whose divisor is $div(f)=4P-4Q$ one should try to verify that $f$ admits a square root in $K(C)$ the function field of $C$.
There is another euristic reason (that maybe can be made precise) that make me think that in general such a couple of points does not exist. Namely if I take a very general smooth quartic $C \subset \mathbb{P}^2$ so that I may suppose that it does not have flexes of order $4$ (equivalently $4P$ does not belong to the canonical system for any $P \in C$); then the $g^1_4$ induced by divisors of the form $4P$ are a $1-$parameter family and I expect that the ramification is $3P+\sum_{i=1}^9P_i$ so that generically the $P_i$'s are distinct and in "singular" cases we have at worst points of multiplicity $2$ apart from $P$ (at least for a generic $C$).
Best Answer
The dimension of the subvariety $Z$ of the moduli space $M_3$ of genus 3 curves that have a pair of points $P \ne Q$ with $4P \sim 4Q$ is 5, so it is a divisor in $M_3$. Indeed, the linear system generated by the divisors $4P$ and $4Q$ defines a morphism $$ f \colon C \to \mathbb{P}^1 $$ which has ramification index 4 at $P$ and $Q$. By Hurwitz formula it follows that $f$ has at most 8 branch points, so the position of these points depend on $8 - 3 = 5$ parameters. The rest of the ramification data is discrete, hence $\dim(Z) \le 5$.
However, the hyperelliptic locus (which also has dimension 5) is not the only component of $Z$. For instance, the curve $$ x^3y + xy^3 + z^4 = 0 $$ has $4P \sim K_C \sim 4Q$, where $P = (1,0,0)$ and $Q = (0,1,0)$, and is not hyperelliptic.