By Andreas Gathmann

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In P2 such lines would meet at infinity, so the intersection would be non-empty then. 10. Every regular function on a complete variety is constant. Proof. Let f : X → A1 be a regular function on a complete variety X. Consider f as a morphism to P1 that does not assume the value ∞. 7. 11. ) Let fi (x0 , . . , xn ), 0 ≤ i ≤ N = n+d n − 1 be the set of all monomials in k[x0 , . . e. of the monomials of the form x00 · · · xnin with i0 + · · · + in = d. Consider the map F : Pn → PN , (x0 : · · · : xn ) → ( f0 : · · · : fN ).

For every f ∈ k[x0 , . . e. f = ∑ f (d) with f (d) homogeneous of degree d for all d. 8. Let I ⊂ k[x0 , . . , xn ] be an ideal. The following are equivalent: (i) I can be generated by homogeneous polynomials. (ii) For every f ∈ I we have f (d) ∈ I for all d. An ideal that satisfies these conditions is called homogeneous. Proof. (i) ⇒ (ii): Let I = ( f1 , . . , fm ) with all fi homogeneous. Then every f ∈ I can be written as f = ∑i ai fi for some ai ∈ k[x0 , . . , xn ] (which need not be homogeneous).

This is obviously not satisfied for our space X. But the analogous definition does not make sense in the Zariski topology, as non-empty open 2. Functions, morphisms, and varieties 29 subsets are never disjoint. Hence we need a different characterization of the geometric concept of “doubled points”. 5. 6. Let X be the complex affine curve X = {(x, y) ∈ C2 ; y2 = (x − 1)(x − 2) · · · (x − 2n)}. 1 that X can (and should) be “compactified” by adding two points at infinity, corresponding to the limit x → ∞ and the two possible values for y.