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If I were to pick just one book to go to, for a search of Civil War information, this one would have to be it.
From Battle, Politics, Leaders, speechs, debates, economics, literature, etc., IT IS ALL IN HERE.
This is one book that everyone would be proud to own. ( As well, as the kind that almost caves your chest in, laying in bed reading--It IS A BIG BOOK.)
There is such a wealth of information, and every thing is solidily backed up with excellent references. It's a fantastic book.
In chapter 1, the main definitions are given and the basic concepts behind schemes outlined. That schemes are more complicated than varieties is readily apparent even in this beginning chapter, where they are thought of as corresponding to the spectrum of a commutative ring with identity. Very elementary exercises are given to help the reader gain confidence in the constructions involved. They authors do have to discuss some sheaf theory, but they show its relevance nicely in this chapter. They also discuss the notion of a fibered product as a generalization of the idea of a preimage of a set under the application of a function and relate it to the construction of the functor of points. The role of the functor of points as reducing schemes to a kind of set theory is brought out beautifully here.
The next chapter gives many examples of schemes, with the first examples being reduced schemes over algebraically closed fields, these being essentially the ordinary varieties of classical algebraic geometry. The authors then give examples of schemes, the local schemes, which are more general than varieties. When departing from the assumption of a field that is not finitely generated, extra points will have to be added to classical varieties. The fact that only one closed point appears is compared to the case of complex manifolds, via the concept of a germ. This is a very helpful comparison, and one that further solidifies the understanding of a scheme in the mind of the reader. The authors give the reader a short peek at the etale topology in one of the examples. Examples are then given where the field is not algebraically closed, generalizing classical number theory, and non-reduced schemes, where nilpotents are present. The chapter ends with examples of arithmetic schemes where the spectra of rings are finitely generated over the integers.
Projective schemes are the subject of Chapter 3, and are defined in terms of graded algebras and invariants of projective schemes embedded in projective space are discussed. The Grasmannian scheme is discussed in detail as an example of a projective scheme. Interestingly, Bezout's theorem, very familiar from elementary algebraic geometry, is generalized here to projective schemes.
Constructions from classical algebraic geometry are generalized to schemes in Chapter 4. The first one discussed is the notion of a flex, which deals (classically) with the locus of tangent lines to a variety. The flexes are defined in terms of the Hessian of the variety, the latter being generalized by the authors to define a scheme of flexes. The notion of blowing up is also generalized to the scheme setting, with the authors motivating the discussion by blowing up the plane. The discussion of blow-ups along non-reduced subschemes of a scheme and blow-ups of arithmetic schemes is fascinating and the presentation is crystal clear. Fano varieties are also generalized to Fano schemes in the chapter. Most of the information about these schemes are contained in the exercises, and some of these need to be worked out for a thorough understanding.
The next chapter is more categorical in nature, and deals with generalizations of the classical Sylvester construction of resultants and discriminants to the scheme setting.
In the last chapter the authors return to the functor of points, and motivate the discussion by asking for a parametrization of families of schemes. The authors show, interestingly, that using the functor of points one can more easily compute geometric information about a scheme than using its equations. They illustrate this for the Zariski tangent space. Then after an overview of Hilbert schemes they close the book by introducing the reader to moduli spaces and a hint of algebraic stacks. No end in sight for this beautiful subject..........
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Wanna learn where to hike? or how to dig a cat hole? or the best place to camp without establishing a new site? Just what the heck should I do with those slimy food orts at the bottom of my cookpot near the top of some peak? Take a read of Soft Paths and you'll find out.
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Simply loaded with photos, drawing and artwork through the ages and for a wide variety of nations, this work covers the origins of heraldry, the development of the College of Arms. It devotes a large section to Scottish Heraldry and the Lyon Court finally ending up with its modern day relevance and usage. They cover the English Coronation Ceremony. Has a superb glossy of terms. The pictures are simply amazing of the Scottish and England's sovereigns' crowns and septers, moving onto to world wide, such as the royalty of Hawaii.
I simply cannot stress how necessary this is to writers, but is wonderful for anyone wanting to know about this topic.
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