By Prof. Dr. Gerd Fischer, Dr. Jens Piontkowski (auth.)

ISBN-10: 3322802175

ISBN-13: 9783322802170

ISBN-10: 3528031387

ISBN-13: 9783528031381

Ruled forms are unions of a kin of linear areas. they're items of algebraic geometry in addition to differential geometry, specifically if the ruling is developable.

This ebook is an advent to either elements, the algebraic and differential one. ranging from very straightforward evidence, the mandatory concepts are built, specifically touching on Grassmannians and primary types in a model appropriate for complicated projective algebraic geometry. eventually, this ends up in contemporary effects at the category of developable governed kinds and evidence approximately tangent and secant varieties.

in comparison to many different subject matters of algebraic geometry, this is often a space simply available to a graduate course.

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**Additional info for Ruled Varieties: An Introduction to Algebraic Differential Geometry**

**Example text**

1jJ(s). If Q is a moving Proof 1) Assume 1jJ is constant and take a basis (vo, ... , Vk) of V vector on U C S, then there are hoI omorphic functions Ai on U such that k k Q=LAiVi' hence Q' = L A; Vi i=O E V, i=O and this implies 1jJ(l) C 1jJ. Conversely, if 1jJ(I) = 1jJ then for every local basis (Qo, ... , Qk) on U we have holomorphic functions A{ on U such that k Qj = L A{ Qi for j = 0, ... ,k. i=O We consider the holomorphic maps " . U ~ ~ <>: N+1, IAk+1

Proposition. Given a compact Riemann suiface S and a holomorphic curve tfr : S ~ G(k, N) of drill d, there exist holomorphic curves tfr(1) : S ~ G(k + d, N) , tfr(l): S ~ G(k - d, N) such that tfr(l)(s) = tfr+(s) and tfr(l)(s) = tfr-(s) for almost every s E tions are dual in the following sense: s. These construc- Proof We take a point a E S and a neighborhood U with a distinguished local basis QO, ... ,Qk as in the Main Lemma. On U \ {a} we have the equalities tfr+ = span {Qo, ... ,Qk. Q~, ...

4, the map The holomorphic curve cp n l{f can be derived from a holomorphic sum by duality, since The sum cp + l{f is called direct if m = k + 1 + 2, which is equivalent to n = 0; we write 1 Grassmannians 44 Using this concept of direct sums, we can show the existence of sufficiently many directrices. Lemma. Given a compact Riemann surface S and two curves 1/1: S --+ f{J: S --+ G(k, N), with f{J C G(l, N) 1/1, there exist directrices f{Jl, ... ·. ,vk, WI, ... , W/-k, ul,· .. - such that the first k + 1 vectors are a base of f{J(0) and the first 1 + 1 vectors are a base of 1/1 (0).

### Ruled Varieties: An Introduction to Algebraic Differential Geometry by Prof. Dr. Gerd Fischer, Dr. Jens Piontkowski (auth.)

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