A Remark on Normal Varieties
H. T. Muhly · Annals of Mathematics · 1941
1. In the terminology of the Italian School an algebraic variety is called if its system of hyperplane sections is complete. 0. Zariski applies the term to an algebraic variety whose associated ring of homogeneous coordinates is integrally closed.' The two concepts are not equivalent. Zariski refers to a variety which satisfies the former condition as in the geometric and to one which satisfies the latter condition as in the arithmetic A variety which is normal in the arithmetic sense is necessarily normal in the geometric sense. Moreover, an r-dimensional variety Vt which is normal in the arithmetic sense has no (r l)-dimehsional singularities. A variety which is normal in the geometric sense need not be normal in the arithmetic sense. For example, a plane quartic of genus two is normal in the geometric sense, but has a double point. A curve may be free from singularities and yet not normal in the arithmetic sense. A rational space quartic illustrates this possibility. The validity of all of these assertions is established in Z. The object of this note is to characterize geometrically those algebraic varieties which are normal in the arithmetic sense. To this end we propose the following theorem: A necessary and sufficient condition that the r-dimnensional algebraic variety Vt be normal in its ambient projective space Pn is that for every integer m the linear system cut out on V, by the hypersurfaces of order m in Pn be complete. In the course of the proof we need the notion of the character of homogeneity of an algebraic variety, introduced by Zariski in the paper Z. Let to * n be the homogeneous coordinates of the general point of a variety Vr in the projective space Pn. The underlying field of constants for Pn is assumed to be algebraically closed and of characteristic zero. We denote this field by K. The integral closure, K[o, P, , I '] of the ring of homogeneous coordinates K[t', , .. , in] in its quotient field 2* is denoted by 6*. Each of the quantities to, t, . , may be assumed to be homogeneous of positive degree.2 Zariski proves that there exist integers 6 with the following property.