By Andreas Gathmann

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Then the union of all lines in Pn intersecting X and Y is a closed subset of Pn . It is called the join J(X,Y ) of X and Y . 8. Recall that a conic is a curve in P2 that can be given as the zero locus of an irreducible homogeneous polynomial f ∈ k[x0 , x1 , x2 ] of degree 2. Show that for any 5 given points P1 , . . , P5 ∈ P2 in general position, there is a unique conic passing through all the Pi . This means: there is a non-empty open subset U ⊂ P2 × · · · × P2 such that there is a unique conic through the Pi whenever (P1 , .

5 (i). 4. There is a completely analogous description of Pn as An with some points added “at infinity”: let P = (a0 : · · · : an ) ∈ Pn be a point. Then we have one of the following cases: (i) a0 = 0. Then P = (1 : α1 : · · · : αn ) with αi = aa0i for all i. The αi are the affine coordinates of P; they are uniquely determined by P and are obtained by setting a0 = 1. So the set of all P with a0 = 0 is just An . e. P = (0 : a1 : · · · : an ), with the ai still defined only up to a common scalar. Obviously, the set of such points is Pn−1 ; the set of all one-dimensional linear subspaces of An .

R and xi = yi for i = 1, . . , n, where x1 , . . , xn , y1 , . . , yn are the coordinates on A2n . 3. e. if ∆(X) is closed in X × X then so is ∆(Y ) in Y ×Y . 5. 3 in the case of the affine line with a doubled origin. So let X1 = X2 = A1 , and let X be the prevariety obtained by glueing X1 to X2 along the identity on A\{0}. 13. As we glue along A1 \{0} to obtain X, it follows that the space X × X contains the point (P, Q) ∈ A1 × A1 • once if P = 0 and Q = 0, • twice if P = 0 and Q = 0, or if P = 0 and Q = 0, • four times if P = 0 and Q = 0.