Intersection Theory Class 13
نویسنده
چکیده
We want to generalize this to cones. Here again is the definition of a cone on a scheme X. Let S = ⊕i≥0S i be a sheaf of graded OX-algebras. Assume OX → S 0 is surjective, S is coherent, and S is generated (as an algebra) by S. Then you can define Proj(S), which has a line bundle O(1). Proj(S) → X is a projective (hence proper) morphism, but it isn’t necessarily flat! (Draw a picture, where the cone has components of different dimension.) Flat morphisms have equidimensional fibers, and cones needn’t have this.
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