Every smooth projective surface X over an algebraically closed field of char p ≥ 7 with h¹(X, O_X) = 2 and p₁(X) = p₂(X) = 1 is birational to an Abelian surface.
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Classifies rational (quasi-)elliptic surfaces with global vector fields in char p ≠ 2, determining fibers, automorphism schemes, moduli, and Jacobian property except for p=3,5.
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Enriques' characterization of Abelian surfaces in positive characteristic
Every smooth projective surface X over an algebraically closed field of char p ≥ 7 with h¹(X, O_X) = 2 and p₁(X) = p₂(X) = 1 is birational to an Abelian surface.
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Rational (quasi-)elliptic surfaces with global vector fields in odd characteristic
Classifies rational (quasi-)elliptic surfaces with global vector fields in char p ≠ 2, determining fibers, automorphism schemes, moduli, and Jacobian property except for p=3,5.