Clifford Theorem for real algebraic curves

In this note, a real algebraic curve X is a smooth proper geometrically integral scheme over R of dimension 1. A closed point P of X will be called a real point if the residue field at P is R, and a non-real point if the residue field at P is C. The set of real points, X(R), will always be assumed to be non empty. It decomposes into finitely many connected components, whose number will be denoted by s. By Harnack’s Theorem we know that s ≤ g + 1, where g is the genus of X. A curve with g + 1− k real connected components is called an (M − k)-curve. The group Div(X) of divisors on X is the free abelian group generated by the closed points of X. If D is a divisor on X, we will denote by O(D) its associated invertible sheaf. The dimension of the space of global sections of this sheaf will be denoted by `(D). Let D ∈ Div(X), since a principal divisor has an even degree on each connected component of X(R) ([8] Lem. 4.1), the number δ(D) (resp. β(D)) of connected components C of X(R) such that the degree of the restriction of D to C is odd (resp even), is an invariant of the linear system |D| associated to D. Let K be the canonical divisor. If `(K − D) = dimH(X,O(D)) > 0, D is said to be special. If not, D is said to be non-special. By Riemann-Roch, if deg(D) > 2g − 2 then D is non-special. Assume D is effective and let d be its degree. If D is non-special then the dimension of the linear system |D| is given by Riemann-Roch. If D is special, then the dimension of the linear system |D| satisfies dim |D| ≤ 1 2 d.