Type II unprojections, Fano threefolds and codimension four constructions

The classification of singular Fano 3-folds remains an open problem in algebraic geometry. The purpose of this thesis is to prove the existence of new families of singular Fano 3-folds, specifically those whose anticanonical embedding is in codimension 4. This is achieved using unprojections. The projection of a scheme Y with coordinate ring k[Y ] := C[x0; : : : ; xn]=IY is a scheme X defined by the coordinate ring k[X] := C[x0; : : : ; xm]=(IY \ C[x0; : : : ; xm]) where m < n. Unprojection is a method of adjoining variables and equations to the ring of X to recover Y . In Chapter 2, we define a new unprojection format allowing us to construct Gorenstein rings of codimension n + 2 from Gorenstein rings of codimension n. Following the naming conventions in the literature, these unprojections are type II and should be considered as type II1 unprojections. We focus on the case where n = 2. This constructs codimension 4 Gorenstein rings which we define explicitly in Chapter 3. In Chapter 4, we use codimension 4 Gorenstein rings to construct and prove the existence of 16 new families of codimension 4 Fano 3-folds. Demonstrably, each family corresponds to a distinct Hilbert series. By using type II1 unprojections as in [33], we construct a second topologically distinct family of codimension 4 Fano 3-folds for these Hilbert series. The Hilbert scheme of these Fano 3-folds, therefore, contains at least 2 components that parametrize distinct Fano 3-folds. In Chapter 5, we consider pre-existing families of codimension 4 Fano 3-folds which are described by [10] but also constructible using our methods.