# What are the possible Clifford functions of a curve?

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 $$C$$?

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.