Clifford index and Clifford dimension

Question

Let $C$ be a smooth projective curve. Let $A\in Pic(C)$. The Clifford index of $A$ is defined as $$Cliff (A)= deg\,A-2(h^0(A)-1).$$ What does this actually measure.

Next the Clifford index of $C$ is defined as $$Cliff(C)=min\{Cliff(A): A\in Pic(C),h^i(A)\geq 2\}.$$ Again the same question, what does this measure and why is there an assumption on global section dimensions.

Finally the Clifford dimension of $C$ is defined as $$min\{h^0(A)-1: A \text{ computes } Cliff(C)\}.$$

What are the motivations for all these definitions. What do these invariants measure. Clarifications will be very helpful.

Answer 1

The Clifford index comes from Clifford's theorem which bounds the how special a linear system can be in terms of it's degree. I think this is interesting and nice in and of itself. In my dotage details are fuzzy, but my recollection is that the definition came about from studying K-3 surfaces. There was a conjecture about the gonality of curves on a given K-3 surfaces. In the end the correct result was proved by Green and Lazarsfeld who showed that this conjecture was false and the correct notion was Clifford index. A big piece of the work is encapsulated in the theorem that on a K-3 surface, the Clifford index of smooth curves in a linear system |L| is constant. I know that there are papers written discussing when the Clifford index is computed by gonality. If you don't make the assumption that L and $ K_X\otimes L^{-1}$ are special then the Clifford index will be 0 and computed by $K_X$.
So the Clifford index of a line bundle measures how special the line bundle is (0 the most special) and the Clifford index of a curve measures how special the curve is in terms of the existence of special linear systems.
A couple of related points. Given the Geometric Riemann-Roch theorem, one can interpet the Clifford index as saying something about the existence of points in special position in the canonical embedding. k < g general points will always span a (k-1) plane, but on a hyperelliptic curve there are 2 points that span a point (the canonical map isn't an embedding). Always speaking in terms of the canonical embedding, a curve of Clifford index one is equivalent to the existence of either a 3 secant line or a 5 pointed plane ( a trigonal curve has a $g^1_3$ or is a plane quintic (has a $g^2_5$).
There is also an interpretation in terms of the bi-canonical embedding (it is always an embedding). Any point in the projective space $P(H^0(K_X^2)^{*} )$ corresponds to an infinitesimal deformation up to scalar. These can be interpreted as maps $\xi \in H^1(T_X) , \xi: H^0(K_C) \to H^1(\mathcal{O}_X)$ and the rank is well defined. Deformations corresponding to the points of X (in the bi-canonical embedding) can easily be shown to be rank 1 (they are called Shiffer deformations). The question is are they the only deformations of rank 1 ? The answer is yes if and only if X has Cliff(X) > 1. Incidentally this is equivalent to the fact that a non-hyperelliptic curve is cut out by quadratic equation in the canonical embedding iff it's Clifford index is greater than 1. This can be generalized. Since the sum of k rank one matrices has rank at most k, one can ask if every element of $P(H^0(K_X^2)^{*} )$ of rank at most k is the sum of k Shiffer deformations. This is true iff the Clifford index of the curve is > k.