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Presented By: Student Algebraic Geometry Seminar - Department of Mathematics

Student Algebraic Geometry: The Cone Theorem and the Classification of Threefold Contractions

Demir Eken

The Cone Theorem describes the cone of effective curves on a projective variety by showing that all K_X-negative directions are generated by extremal rays spanned by rational curves. We will discuss how these rays correspond to specific geometric contractions—divisorial contractions, fiber spaces, and flips—which drive the Minimal Model Program. Finally, we use a projective threefold example to show how rational curves explicitly determine the possible contractions.

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