## Algebraic Geometry Iv Linear Algebraic Groups Invariant by A.N. Parshin

By A.N. Parshin

Two contributions on heavily comparable matters: the speculation of linear algebraic teams and invariant idea, by means of recognized specialists within the fields. The ebook could be very worthy as a reference and learn consultant to graduate scholars and researchers in arithmetic and theoretical physics.

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**Extra resources for Algebraic Geometry Iv Linear Algebraic Groups Invariant Theory**

**Example text**

E. e. l(D − 2P ) = l(D) − 2 for any point P ∈ C. 11. P ROPOSITION . Every morphism φ : C → Pr is given by a base-point-free linear system (possibly incomplete) as long as φ(C) is not contained in a projective subspace of Pr (in which case we can just switch from Pr to Ps for s < r). Proof. Indeed, φ is obtained by choosing rational functions f0 , . . , fr ∈ k(C). Consider their divisors (f0 ), . . , (fr ) and let D be the smallest effective divisor such that (fi )+D is effective for every i. Then of course every fi ∈ L(D) and D is base-point-free (otherwise it’s not the smallest).

Notice that in this case Aut(C, P ) modulo the hyperelliptic involution acts on P1 by permuting branch points. In fact, λ is simply the cross-ratio: (p4 − p1 ) (p2 − p3 ) , (p2 − p1 ) (p4 − p3 ) but branch points are not ordered, so we have an action of S4 on possible cross-ratios. However, it is easy to see that the Klein’s four-group V does not change the cross-ratio. 5) Special values of λ correspond to cases when some of the numbers in this list are equal. For example, λ = 1/λ implies λ = −1 and the list of possible πi cross-ratios boils down to −1, 2, 1/2 and λ = 1/(1 − λ) implies λ = e 3 , −πi πi in which case the only possible cross-ratios are e 3 and e 3 .

2. D EFINITION . A morphism f : X → Y of algebraic varieties is called smooth if it is flat, has reduced fibers, and every fiber is non-singular. e. a smooth proper morphism with a section A = σ(B) such that all fibers are elliptic curves. We would like to write down Weierstrass equation of X, possibly after shrinking the base B. If B = pt, the argument will be identical to what we have seen so far, using linear systems L(E, kP ) = H 0 (E, OE (kP )). Recall that if F is a sheaf of Abelian groups on an algebraic variety X then we can define higher cohomology groups H k (X, F) in addition to the group H 0 (X, F) of global sections.