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 d to one of degree d − 1, 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 BuchsbaumEisenbud 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; see also The theoremLet X = Spec O X be a Gorenstein local scheme and I ⊂ O X an ideal sheaf defining a subscheme D = V (I) ⊂ X that is also Gorenstein and has codimension 1 in X. We assume that all schemes are Noetherian. We do not assume anything else about the singularities of X and D, although an important case in applications is when X is normal and D a Weil divisor.The adjunction formula (compare Reid [R], Appendix to Section 2) gives. * The first author thanks the Greek State Scholarships Foundation for support. † We both thank Kyoto University, RIMS for generous support and hospitality.

Orbifold Riemann--Roch for threefolds with an application to Calabi--Yau geometry

“…In answer to the first question, it is certainly possible that our examples are birational to one of the gigantic number of toric complete intersection Calabi-Yau threefolds [5,6,16]. However, our projective descriptions, in relatively simple ambient varieties such as weighted Grassmannians [11] and the "universal Type I unprojection" [18], are new. Such descriptions are quite pretty in themselves, and can also be used for various purposes such as computing their Hodge numbers and studying mirror symmetry phenomena (details to follow).…”

“…The existence of X can be proved by the Type I unprojection [1,18] starting from a codimension 3 variety Y ⊂ P 6 (1, 1, 2, 3, 3, 3, 3) defined by a set of Pfaffians as in Example 3.4. We omit the details.…”

The Cramer varieties Cr(r,r+s,s)

“…Reid and Corti [1] studied weighted analogs of the homogeneous spaces such as Grassmannian Gr (2,5) and the Orthogonal Grassmanian OGr(5, 10) and how to use these as weighted projective constructions.…”

“…Understanding this case led them to the more general notion of Gorenstein unprojection. For r = 4 and s = 1, these varieties in terms of equations are unprojections and have been studied by Papadakis and Reid, see [4,5]. In fact for s = 1 the variety is a single unprojection because all x i = 0 is a codimension (r + 1) complete intersection D and all m ij x j = 0 is a codimension r complete intersection X containing D. So Kustin-Miller unprojection applies to give ω as an unprojection variable with the second set of equations as unprojection equations.…”

Birational modifications of surfaces via unprojections

“…As explained in [PR,Section 2.3], the prototype and easiest example of an unprojection is the Castelnuovo blow-down of a rational (−1)-curve lying on a smooth cubic surface in P 3 as the inverse of a projection from a del Pezzo surface of degree 4 in P 4 , which also explains the name unprojection.…”