Normal varieties and normalization

In this entire section, we assume that A is an algebraically closed field. 

Definition 1. Let X be a variety over k. Then a point x 6 X is called a normal point of X, or X is said to be normal at z if the ring ox is integrally closed in its quotient field k(X). X is normal if it is normal at every point. 

for example, that a factorial variety X (i.e., gx always a UFD) is normal, since all UFD s are integrally closed in their quotient fields. In particular, nonsingular varieties are normal. If a variety X is normal, its affine coordinate rings are integrally closed in k(X) too. In fact, an intersection of integrally closed rings is integrally closed and for any domain R, 

Structure theorem of integrally closed noetherian rings. If R is a noethe-rian integral domain, then 

Proposition 1. Let X be a normal variety and let S C X be its singular locus (i.e., the set of singular points). Then 

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Amorphous sulphur is a metastable variety and is insoluble in carbon disulphide. Its change to the stable rhombic form is temperature dependant. At normal ambient temperatures, the amorphous form is virtually stable requiring many years to convert. At 100 °C, the change occurs in about one hour. At about 110 °C the change will occur in about 20 minutes. Care must be taken to ensure that processing temperatures do not exceed 100 °C otherwise the advantages of its use will be quickly lost. 

Chlorophyll content of food systems is normally expressed based on a reference system (e.g., wet or dry weight basis). While wet weight chlorophyll content is often reported, it should be considered in the context of the preparation s total moisture, as water content may potentially vary between varieties and preparations. Therefore, these values should be accompanied by moisture content of the tissue in order to account for this innate variability. Alternatively, chlorophy 11 content expressed on a dry weight basis can easily be compared across varieties and preparations. While dry weight content is preferred, it requires a drying step (preferably freeze drying) prior to analysis,... 

The essential and normal components of wine are the same in almost all varieties and types the respective proportions alone vary in different kinds of wine, which may be grouped in three principal categories, namely, ordinary wine, sweet wine and vins de coupage (used for improving weak wines). 

In another study, 24 healthy, normal-weight men with normal cholesterol levels were fed diets that contained 300 mg or 800 mg of cholesterol per day in ten-day periods, and in a 21-day period, less than 50 mg per day of cholesterol. Subsequently, 8 g, 16 g, or 25 g of olestra were added to each of the diets in ten-day periods. The diets were of the typical American variety and contained approximately 20% of their calories as protein, 40% as carbohydrates, and 40% as fat. Care was taken so that there were no significant reductions in weight. Adding olestra to the diet lowered both total and LDL cholesterol at all levels of cholesterol feeding (57). 

Initially recorded with rare paraproteins (B23), mainly of the IgA class, this has also been associated with IgG and IgM. Usually the total serum cholesterol is raised, but in one case it has been within normal limits curiously it is subnormal in most patients with IgA-myelomatosis without xanthoma (S7). The xanthomata are typically of the soft eruptive variety and contain complexes of the paraprotein and /8-Iipo-protein. Beaumont (B8) has collected evidence suggesting that the paraproteins are antibodies to the 8-apoprotein. Occasionally excess complexes can result in viscosity syndrome (Section 7.5.6). If a lipid stain is used, the paraprotein band is positive. In such cases, regrettably. Potter has been unable to relate the antibody activity to phosphoryl-choline (P13). Cytotoxic treatment can reduce the serum levels of lipid and paraprotein. 

This phenomenon would be of little interest here were it not for some experiments conducted by an American scientist named Warmke during World War II. Treating marijuana plants with colchicine, he obtained a number of triploid and tetraploid and tetraploid plants had about the same average activity (though individual plants varied by a factor of eight) which was about twice chat of the normal diploid strains. He used 26 different plants of each variety and his data were statistically significant. Unfortunately, he worked with an acetone extract of the plants and his assay animals were small fish, making it doubtful whether his data have any relation to cannabinoid content. 

Definition 3. Let X be a variety and let L be a finite algebraic extension of k(X). A normalization of X in L is a normal variety Y with function field k(Y) = L, plus a finite surjective morphism ... 

Using Kummer varieties and suitable normalized coordinate systems one can define the Schottky loci Sg and [Pg.289]

The classification can be extended to amoeboid, dendritic, or dotlike types, but these terms are self-descriptive and need for alphabetic designations appears unnecessary. Thus one might symbolize a belite grain as Type AE, meaning that it has normal lamellae and they extend into the matrix or Type CF, indicating a complex internal structure and satellite crystals, etc. Designations of polymorph variety in the writer s classification are not intended, but polymorph variety and abundance are of major significance. 

The maize kernel contains two main types of protein. Zein, occurring in the endosperm, is quantitatively the more important, but this protein is deficient in the indispensable amino acids tryptophan and lysine (see Fig. 22.2). The other protein, maize glutelin, occurring in lesser amounts in the endosperm and also in the germ, is a better source of these two amino acids. Varieties of maize have been produced with amino acid components different from those present in normal maize. One such variety is Opaque-2, which has a high lysine content. The difference between this variety and normal maize is primarily attributed to the zein/glutelin ratio. 

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