Let $C$ be some smooth proper curve of genus $g$ over an algebraically closed field $k$. In order to understand special divisors on C one may consider the following function c(r), which I will call Clifford's function. It assigns to a number $0 \leq r \leq g-1$ the (minimal degree of a divisor such that r(D) = dim|D| = r) - 2r. Clifford's theorem states that $c(r) \geq 0$. Looking at a $D \in g^1_d$ we notice that $r(nD) \geq n$ and hence $c(r) \leq rc(1)$. From this I arrive at the guess that $c$ is always concave. In fact I guess that there exists some generic Clifford function $C_{max}$ and that we can characterize all Clifford functions as concave functions $0 \leq c \leq C_{max}$.

In my searches I have only been able to find results on special values of the Clifford function (for instance $c(1)$ corresponds to the gonality of a curve), but none on this function as an object. Is it possible to characterize all functions occurring as the Clifford function of some curve $C$?


1 Answer 1


This is a partial answer concerning the existence of such a $C_{max}$ and its concavity. The Brill-Noether theorem states that on a generic curve a $g^r_d$ exists iff $g-(r+1)(g-d+r) \geq 0$. This is equivalent to $d \geq g+r-\frac{g}{r+1}$. It follows that the generic Clifford function is $c(r) = g-r-\lfloor\frac{g}{r+1}\rfloor$ which again by Brill-Noether theory is the maximal such function.

It remains to show that this function is concave which is equivalent to showing that $f(r) = \lfloor\frac{g}{r+1}\rfloor$ is concave within the range. In fact it is enough to show that $f(r-1) + f(r+1) \geq 2 f(r)$ for integers $1 \leq r \leq g-2$. Setting $h(r) = \frac{g}{r+1}$, it is enough to show $h(r-1) + h(r+1) -2 \geq 2h(r)$ for such $r$. This is an easy calculation.

To continue it is probably helpful to understand the basics of the intersection of Brill-Noether loci in the moduli space of curves.


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