Non-K3 Weierstrass numerical semigroups

We generalize the result of Reid (J Lond Math Soc 13:454-458, 1976), namely, we prove that a curve of genus >= g2+4g+6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\geqq g<^>2+4g+6$$\end{document} having a double cover of a hyperelliptic curve of genus g >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\geqq 2$$\end{document} 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.