On finite generation of symbolic Rees rings of space monomial curves and existence of negative curves

In this paper, we shall study finite generation of symbolic Rees rings of the defining ideal of the space monomial curves $(t^a, t^b, t^c)$ for pairwise coprime integers $a$, $b$, $c$ such that $(a,b,c) \neq (1,1,1)$. If such a ring is not finitely generated over a base field, then it is a counterexample to the Hilbert's fourteenth problem. Finite generation of such rings is deeply related to existence of negative curves on certain normal projective surfaces. We study a sufficient condition (Definition 3.6) for existence of a negative curve. Using it, we prove that, in the case of $(a+b+c)^2 > abc$, a negative curve exists. Using a computer, we shall show that there exist examples in which this sufficient condition is not satisfied.Comment: In the previous version, there was a serious mistake in the last sectio

Almost Gorenstein rings

The notion of almost Gorenstein ring given by Barucci and Fr{\"o}berg \cite{BF} in the case where the local rings are analytically unramified is generalized, so that it works well also in the case where the rings are analytically ramified. As a sequel, the problem of when the endomorphism algebra \m : \m of \m is a Gorenstein ring is solved in full generality, where \m denotes the maximal ideal in a given Cohen-Macaulay local ring of dimension one. Characterizations of almost Gorenstein rings are given in connection with the principle of idealization. Examples are explored

The Aligned SU(5)×U(1)2SU(5) \times U(1)^2 Model

In Calabi-Yau string compactification, it is pointed out that there exists a new type of $SU(5) \times U(1)^2$ model (the aligned $SU(5) \times U(1)^2$ model) in which the $SU(5)$ differs from the standard $SU(5)$ and also from the flipped $SU(5)$. With the aid of the discrete symmetry suggested from Gepner model, we construct a simple and phenomenologically interesting three-generation model with the aligned $SU(5) \times U(1)^2$ gauge symmetry. The triplet-doublet splitting problem can be solved. It is also found that there is a realistic solution for solar neutrino problem and for the $\mu$-problem. At low energies this model is in accord with the minimal supersymmetric standard model except for the existence of singlet fields with masses of $O(1)$TeV.Comment: LaTeX 24 page