Research on algebraic varieties with an algebraic group action

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(1) Research on the moduli space of pointed algebraic curves of low genus. We have studied the moduli space M(g,1 ; N) of pointed algebraic curves (X,P) of genus g with a given semigroup N at the point P, and have got a theorem saying that, if the genus g is less than or equal to 6 and the number of generators of the semigroup N is less than or equal to 4, then the moduli space M(g,1 ; N) is an irreducible rational variety. This theorem is contained in the paper titled "On the moduli space of pointed algebraic curves of low genus II - rationality -", which is to appear in Tokyo J. Math. 31(2008).(2) Research on the invariant rings of the finite subgroups of the special linear group SL(4,C) of degree 4. We have been computing the generators and relations of the invariant rings of the finite primitive subgroup of SL(4,C), which are 30 of them in all. So far, we have successfully got the generators and relations of about 15 of them. The remaining groups have big order and the direct computation is difficult. We are now seeking for a method for computation for those subgroups with big order.(3) Research on the toric varieties and the combinatorics. We have studied the complete linear systems on the complete toric Gorenstein del Pezzo surfaces, and determined the unique linear system of minimal dimension and degree. As an application, we can compute the defining equations of the image of the minimal dimensional embedding and also determine if a complete toric Gorenstein del Pezzo surface is a global complete intersection or not.