Classification in Small Dimensions
Fano varieties in dimensions 1 are isomorphic to the projective line.
In dimension 2 they are del Pezzo surfaces and are isomorphic to either or to the projective plane blown up in at most 8 general points, and in particular are again all rational.
In dimension 3 there are non-rational examples. Iskovskih classified the Fano 3-folds with second Betti number 1 into 17 classes, and Mori & Mukai (1981) classified the ones with second Betti number at least 2, finding 88 deformation classes.
Read more about this topic: Fano Variety
Famous quotes containing the words small and/or dimensions:
“Do but consider this small dust, here running in the glass,
By atoms moved.
Could you believe that this the body was
Of one that loved?
And in his mistress’ flame playing like a fly,
Turned to cinders by her eye?
Yes, and in death as life unblest,
To have’t expressed,
Even ashes of lovers find no rest.”
—Ben Jonson (1572–1637)
“Why is it that many contemporary male thinkers, especially men of color, repudiate the imperialist legacy of Columbus but affirm dimensions of that legacy by their refusal to repudiate patriarchy?”
—bell hooks (b. c. 1955)