Algebraic curves through commutative semigroups and its application to toric varieties

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We are interested in the following two semigroups. One is the semigroup whose element is the order of a pole of a rational function with only one pole at a given fixed point on an algebraic curve. This semigroup is called the Weierstrass semigroup of the point. This semigroup becomes a numerical semigroup. Another is the Weierstrass semigroup of the pair of two points which is defined in a similar way to the case of one point. This research is devoted to the following :1. Study on the existence or the construction of a pointed curve whose Weierstrass semigroup is a given numerical semigroup.2. The description of the Weierstrass semigroup of the pair of ramification points of a covering of the projective line.3. Research into affine toric varieties constructed from the Weierstrass semigroup of a point.For the first case we construct a double covering of a hyperelliptic curve which is ramified over a Weierstrass point and investigate the Weierstrass semigroup of the ramification point. We can show that every 4-semigroup, i.e., a numerical semigroup whose minimum positive element is four, is derived from such a construction.For the second case we describe the Weierstrass semigroup of the pair of two total ramification points on a cyclic covering of the projective line with prime degree.The third case corresponds to the application to toric varieties. We study the affine toric varieties constructed from a 6-semigroup or a 7-semigroup generated by 4 elements.In the future we will study the existence and the construction of a pointed curve whose Weierstrass semigroup is of genus 8 or 9 for the first case and a double covering of a hyperelliptic curve for the second case and 6-semigroups, 7-semigroups generated by 5 elements, 2-dimensional affine toric varieties for the third case.