Double Coverings of Curves and Non-Weierstrass Semigroups

Abstract

We give a new method of constructing a non-Weierstrass semigroup H, which means that there is no smooth projective pointed curve over an algebracally closed field of characteristic 0 whose Weierstrass semigroup is H. This method depends on a description of a pointed smooth projective curve such that there exists a double covering of the curve ramified over the point with a certain condition on the genus of the covering curve. Using this we find non-Weierstrass semigroups whose minimum positive integers are 8 and 12, respectively.

Key Words:

• Double coverings of curves
• Non-Weierstrass semigroups
• Symmetric numerical semigroups
• 8-semigroups
• 12-semigroups

2010 Mathematics Subject Classification:

• 14H55
• 14H30
• 20M14

ACKNOWLEDGMENT

The author is partially supported by Grant-in-Aid for Scientific Research (21540052), Japan Society for the Promotion Science.

Notes

Communicated by S. Kleiman.