Algebraic Geometry

Algebraic geometry studies spaces cut out by polynomial equations and translates between their geometric and algebraic invariants.

The largest field, varieties and projective geometry, includes Bézout's theorem for \(n\) hypersurfaces in \(\mathbb{P}^n\) over an algebraically closed field that meet in finitely many points and Seshadri constants of ten points in the plane, whose generic value Nagata's conjecture predicts to be \(1/\sqrt{10}\). For elliptic curves over a field with \(q\) elements, the Hasse bound \((q + 1 - N)^2 \le 4q\) limits the number \(N\) of points and Waterhouse's theorem says which integers occur as \(q + 1 - N\). Algebraic curves come with the Abel–Jacobi map into the Jacobian and the uniform bound of Dimitrov, Gao and Habegger on the rational points of curves of genus at least \(2\), and arithmetic geometry has Faltings's theorem and the Weil conjectures over finite fields.

• Schemes and morphisms
• Varieties and projective geometry
• Algebraic curves
• Elliptic curves
• Arithmetic geometry

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