Get Algebraic Surfaces PDF

By Oscar Zariski

ISBN-10: 0387053352

ISBN-13: 9780387053356

The most objective of this ebook is to provide a very algebraic method of the Enriques¿ class of gentle projective surfaces outlined over an algebraically closed box of arbitrary attribute. This algebraic technique is among the novelties of this ebook one of the different sleek textbooks dedicated to this topic. chapters on floor singularities also are integrated. The ebook could be worthy as a textbook for a graduate direction on surfaces, for researchers or graduate scholars in algebraic geometry, in addition to these mathematicians operating in algebraic geometry or comparable fields"

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Example text

This ensures that the construction can be continued. The observation above on val(cl+1 j ) implies kl+1 ≤ rl+1 ≤ kl . Since n is finite, the value of kl can only drop a finite number of times. Hence there exist k ∈ {1, . . , n} and m ∈ N such that kl = k for all l ≥ m. This means that rl = k for all l > m, so fl = µl (x − λl )k for all l > m, and some µl ∈ k. Let Nl be such that clj ∈ k((t1/Nl )) for 0 ≤ j ≤ n. 2), we can take Nl+1 to be the least common multiple of Nl and the denominator of wl .

Any generator of that kernel translates into a scalar multiple of the polynomial f (x, y). From this example we see that the implicit equation f (x, y) can be recovered using (numerical) linear algebra from the Newton polytope Newt(f ), but it also suggests that the matrices in the resulting systems of linear equations tend to be dense and ill-conditioned. It is hence a rather non-trivial computational problem to solve the equations when f (x, y) has thousands of terms. However, from a geometric perspective it makes sense to consider the implicitization problem solved once the Newton polytope has been found.

2), we can take Nl+1 to be the least common multiple of Nl and the denominator of wl . Let yl = lj=0 λj tw0 +···+wj ∈ k((t1/Nl )). We claim that Nl+1 = Nl works for l > m. Indeed, we have wl+1 = val(cl0 )/k, so it suffices to show val(cl0 ) ∈ Nkl Z for l > m. This follows from the fact that fl is a pure power, so val(clj ) = (k − j) val(clk−1 ) for 0 ≤ j ≤ k, and in particular val(clk−1 ) = 1/k val(cj0 ) ∈ N1l Z. Thus there is an N for which yl ∈ k((t1/N )) for all l, and so the limit λj tw0 +···+wj y = lies in k((t1/N )).

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Algebraic Surfaces by Oscar Zariski

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