# On a smooth curve $C$, when is $K_C \sim_\mathbb{Q} (2g-2)P$?

Let $C$ be a smooth curve of genus $g$ over $\mathbb{C}$. I am interested in the following property:

There exists a point $P \in C$ such that $K_C \sim_\mathbb{Q} (2g-2)P$. Equivalently, $K_C - (2g-2)P$ is torsion in $\mathrm{Pic}^0(C)$.

My question is the following. How special is this property? More precisely, fix a genus $g \geq 3$. Are the curves in $\mathcal{M}_g$ with this property a (countable) union of closed subsets?

Some special kinds of curves (e.g. hyperelliptic, or totally branched over $\mathbb{P}^1$) have this property, but it seems to me that it is something quite strong to require. Also, in view of results about torsion packets on curves (see for instance http://people.math.gatech.edu/~mbaker/pdf/packets-final.pdf), the point $P$ above needs to be essentially unique.

• The condition that $n K_C = n (2g-2) P$ in $Pic^0$ is clearly a closed condition on $(C,P) \in M_{g,1}$, and the projection from $M_{g,1}$ to $M_g$ is proper, so the set of such $C$ is clearly closed (for each $n$). The nontrivial question is to show that it is not all of $M_g$, and perhaps to estimate its dimension. Mar 9, 2018 at 18:46
• A naive dimension count suggests dimension $2g-2$ (condition on $M_{g,1}$ is naively codimension $g$, and $\dim M_{g,1} = (3g-3)+1$, no reason to think the projection to $M_g$ drops dimension). However, the hyperelliptic locus has this property and has dimension $2g-1$, so there might be other higher dimensional components as well. Mar 9, 2018 at 18:49
• The Baker-Poonen result on torsion packets proves that the projection $M_{g,1} \to M_g$ is basically one-to-one on this locus, so we can concentrate on understanding the locus in $M_{g,1}$. Mar 9, 2018 at 19:14
• The point $P$ is not necessarily unique. On the Fermat curve $X^n+Y^n+Z^n=0$ there are $3n$ such points, namely the points $(\alpha ,1,0)$, $(\alpha ,0,1)$ and $(0,1,\alpha)$ with $\alpha ^n=-1$.
– abx
Mar 9, 2018 at 20:50
• @DavidESpeyer Can you say something more about the codimension count of the condition in $M_{g,1}$? How do you estimate it to be $g$? Mar 14, 2018 at 20:35