Over an algebraically closed field, radical ideals of \(k[x_1,\ldots,x_n]\) correspond to closed subsets of \(\AA^n\). \(\PP^1\times\PP^1\) is projective by the Segre embedding, the complement of a curve in \(\PP^2\) is affine, and two plane curves meeting in infinitely many points share a common component.
For a projective variety \(X\) of dimension \(n\) and degree \(d\), the Hilbert polynomial is \(P_X(r)=\tfrac{d}{n!}r^n+\cdots\), and for a curve \(P_X(0)=1-p_a\).
The dictionary, radical ideals against closed sets, and irreducible decomposition.
Affine or projective, criteria for affineness, projectivity, and completeness.
Dimension and degree, and the Hilbert polynomial.
Smooth and singular, the Jacobian and regular-local-ring criteria, Bertini’s theorem, normality, and singular loci.
Regular functions, the rings \(A(X)\), \(\OO_{X,p}\), and \(k(X)\), and quasi-affine varieties.
Blowups, the exceptional divisor, the canonical class of a blowup, and Castelnuovo contraction.
Varieties by canonical class, Fano and Calabi–Yau varieties, and the canonical class of a hypersurface.