On the classification of non-aCM curves on quintic surfaces in ℙ^3

In this paper, a curve is any projective scheme of pure dimension one. It is well known that the arithmetic genus and the degree of an aCM curve D in ℙ^3 are computed by the h -vector of D . However, for a given curve D in ℙ^3 , the two aforementioned invariants of D do not tell us whether D is aCM or not. If D is an aCM curve on a smooth surface X in ℙ^3 , any member of the linear system |D+lC| is also aCM for each non-negative integer l , where C is a hyperplane section of X . By a previous work, if a non-zero effective divisor D of degree d and arithmetic genus g on a smooth quintic surface X in ℙ^3 is aCM and satisfies the condition h^0(𝒪_X(D-C))=0 , then 0≤ d+1-g≤ 4 . In this paper, we classify non-aCM effective divisors on smooth quintic surfaces in ℙ^3 of degree d and arithmetic genus g such that 0≤ d+1-g≤ 4 , and give several examples of them.