Such complete intersections have important applications in geometry and number theory, because they typically admit rational points, an elementary case of which is the Chevalley–Warning theorem.
Fano varieties provide an abstract generalization of these basic examples for which rationality questions are often still tractable.
For a Fano variety X over the complex numbers, the Kodaira vanishing theorem implies that the sheaf cohomology groups
cases of this vanishing statement also tell us that the first Chern class induces an isomorphism
By Yau's solution of the Calabi conjecture, a smooth complex variety admits Kähler metrics of positive Ricci curvature if and only if it is Fano.
Campana and Kollár–Miyaoka–Mori showed that a smooth Fano variety over an algebraically closed field is rationally chain connected; that is, any two closed points can be connected by a chain of rational curves.
[1] Kollár–Miyaoka–Mori also showed that the smooth Fano varieties of a given dimension over an algebraically closed field of characteristic zero form a bounded family, meaning that they are classified by the points of finitely many algebraic varieties.
The following discussion concerns smooth Fano varieties over the complex numbers.
Every del Pezzo surface is isomorphic to either P1 × P1 or to the projective plane blown up in at most eight points, which must be in general position.
In dimension 3, there are smooth complex Fano varieties which are not rational, for example cubic 3-folds in P4 (by Clemens - Griffiths) and quartic 3-folds in P4 (by Iskovskikh - Manin).
Iskovskih (1977, 1978, 1979) classified the smooth Fano 3-folds with second Betti number 1 into 17 classes, and Mori & Mukai (1981) classified the smooth ones with second Betti number at least 2, finding 88 deformation classes.
A detailed summary of the classification of smooth Fano 3-folds is given in Iskovskikh & Prokhorov (1999).