(Fano, Calabi--Yau, and general type)
Let \(X\) be a smooth complete variety with canonical divisor \(K_X\).
\(X\) is Fano if \(-K_X\) is ample.
\(X\) is Calabi–Yau if \(K_X \sim_\QQ 0\); the stricter convention demands \(\omega_X \cong \OO_X\) together with \(h^j(\OO_X) = 0\) for \(1 \leq j \leq \dim X - 1\).
\(X\) is of general type if \(K_X\) is big.
A del Pezzo surface is a Fano variety of dimension two.
A smooth projective curve of genus \(g\) is Fano if and only if \(g = 0\), Calabi–Yau if and only if \(g = 1\), and of general type if and only if \(g \geq 2\), since \(\deg K=2g-2\). A complete variety with an ample divisor is projective, so every Fano variety is projective.
The two conventions for Calabi–Yau are not equivalent: abelian varieties have \(K_X=0\) but do not satisfy the intermediate-cohomology vanishing, so they satisfy the first convention and not the second. The del Pezzo surfaces are \(\PP^1\times\PP^1\) and \(\PP^2\) blown up at \(r\le8\) general points, with \(K^2=9-r\); the cubic surface is the case \(r=6\) and contains \(27\) lines.
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FE-VARADJHypersurfaces in $\PP^n$, classified by adjunctionuses