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Algebraic Geometry by Andreas Gathmann

By Andreas Gathmann

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Fukaya Categories and Picard-Lefschetz Theory

The relevant items within the booklet are Lagrangian submanifolds and their invariants, similar to Floer homology and its multiplicative constructions, which jointly represent the Fukaya class. The proper facets of pseudo-holomorphic curve conception are coated in a few aspect, and there's additionally a self-contained account of the mandatory homological algebra.

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Xn ) ∈ X ⇒ (λx0 , . . , λxn ) ∈ X. If X ⊂ Pn is a projective algebraic set, then C(X) := {(x0 , . . , xn ) | (x0 : · · · : xn ) ∈ X} ∪ {0} is called the cone over X (obviously this is a cone). 2. In other words, a cone is an algebraic set in An+1 that can be written as a (usually infinite) union of lines through the origin. The cone over a projective algebraic set X ⊂ Pn is the inverse image of X under the projection map An+1 \{0} → (An+1 \{0})/(k\{0}) = Pn , together with the origin. 3. ) Note that C(X) contains the two lines L1 and L2 , which correspond to “points at infinity” of the projective space P2 .

For every open subset U ⊂ Y define OY (U) to be the ring of k-valued functions f on U with the following property: for every point P ∈ Y there is a neighborhood V of P in X and a section F ∈ OX (V ) such that f coincides with F on U. (i) Show that the rings OY (U) together with the obvious restriction maps define a sheaf OY on Y . (ii) Show that (Y, OY ) is a prevariety. 9. Let X be a prevariety. Consider pairs (U, f ) where U is an open subset of X and f ∈ OX (U) a regular function on U. We call two such pairs (U, f ) and (U , f ) equivalent if there is an open subset V in X with V ⊂ U ∩U such that f |U = f |U .

This follows easily from the fact that Z( f ) = Z(x0d f , . . , xnd f ) for all homogeneous polynomials f and every d ≥ 0. 12. Let L ⊂ An+1 be a linear subspace of dimension k + 1; it can be given by n − k linear equations in the coordinates of An+1 . e. the subspace of Pn given by the same n − k equations (now considered as equations in the homogeneous coordinates on Pn ) is called a linear subspace of Pn of dimension k. Once we have given projective varieties the structure of varieties, we will see that a linear subspace of Pn of dimension k is isomorphic to Pk .

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