We generalize the result of Reid (J Lond Math Soc 13:454–458, 1976), namely, we prove that a curve of genus \(\geqq g^2+4g+6\) having a double cover of a hyperelliptic curve of genus \(g\geqq 2\) does not lie as a non-singular curve on any K3 surface. Applying this result we construct non-K3 Weierstrass numerical semigroups. A numerical semigroup H is said to be Weierstrass if there exists a pointed non-singular curve (C, P) such that H consists of non-negative integers which are the pole orders at P of a rational function on C having a pole only at P. We call the numerical semigroup K3 if we can take the curve C as a curve on some K3 surface. A non-K3 numerical semigroup means that it cannot be attained by a pointed non-singular curve on any K3 surface. We also give infinite sequences of non-K3 Weierstrass numerical semigroups.
