Kustin–Miller unprojection without complexes

Abstract

Gorenstein projection plays a key role in birational geometry; the typical example is the linear projection of a del Pezzo surface of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to one of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d minus 1"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , but variations on the same idea provide many of the classical and modern birational links between Fano 3-folds. The inverse operation is the Kustin–Miller unprojection theorem, which constructs “more complicated” Gorenstein rings starting from “less complicated” ones (increasing the codimension by 1). We give a clean statement and proof of their theorem, using the adjunction formula for the dualising sheaf in place of their complexes and Buchsbaum–Eisenbud exactness criterion. Our methods are scheme theoretic and work without any mention of the ambient space. They are thus not restricted to the local situation, and are well adapted to generalisations. Section 2 contains examples, and discusses briefly the applications to graded rings and birational geometry that motivate this study.