Fano variety

In algebraic geometry, a Fano variety, introduced by Gino Fano in , is a complete variety X whose anticanonical bundle KX* is ample. In this definition, one could assume that X is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. Recently techniques in differential geometry have been applied to the study of Fano varieties over the complex numbers, and success has been found in constructing moduli spaces of Fano varieties and proving the existence of Kähler–Einstein metrics on them through the study of K-stability of Fano varieties. The fundamental example of Fano varieties are the projective spaces: the anticanonical line bundle of Pn over a field k is O(n+1), which is very ample (over the complex numbers, its curvature is n+1 times the Fubini–Study symplectic form). Let D be a smooth codimension-1 subvariety in Pn. The adjunction formula implies that KD = (KX + D)|D = (−(n+1)H + deg(D)H)|D, where H is the class of a hyperplane. The hypersurface D is therefore Fano if and only if deg(D) < n+1. More generally, a smooth complete intersection of hypersurfaces in n-dimensional projective space is Fano if and only if the sum of their degrees is at most n. Weighted projective space P(a0,...,an) is a singular (klt) Fano variety. This is the projective scheme associated to a graded polynomial ring whose generators have degrees a0,...,an. If this is well formed, in the sense that no n of the numbers a have a common factor greater than 1, then any complete intersection of hypersurfaces such that the sum of their degrees is less than a0+...+an is a Fano variety. Every projective variety in characteristic zero that is homogeneous under a linear algebraic group is Fano. The existence of some ample line bundle on X is equivalent to X being a projective variety, so a Fano variety is always projective. For a Fano variety X over the complex numbers, the Kodaira vanishing theorem implies that the sheaf cohomology groups of the structure sheaf vanish for .

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Minimal model program

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Birational geometry

In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles. A rational map from one variety (understood to be irreducible) to another variety , written as a dashed arrow X Y, is defined as a morphism from a nonempty open subset to .