AI时间线birational geometry的历史时间线
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Birational geometry is a field of algebraic geometry that studies birational maps between algebraic varieties, aiming to classify varieties up to birational equivalence. Key developments include the resolution of singularities, minimal model program, and the recent advances in the theory of log canonical models and moduli spaces.
19th century
Foundations of birational geometry laid by Riemann, Clebsch, and others, using rational maps to study algebraic curves and surfaces.
1890s
Italian school of algebraic geometry (Castelnuovo, Enriques, Severi) develops classification of algebraic surfaces up to birational equivalence, introducing concepts like minimal models for surfaces.
1940s-1950s
Zariski and others develop formal foundations for birational geometry, including the notion of birational equivalence and the resolution of singularities for surfaces and threefolds.
1964
Hironaka proves the resolution of singularities in characteristic zero, a crucial tool for birational geometry.
1970s-1980s
Mori's proof of the cone theorem and the existence of flips for threefolds, laying the groundwork for the minimal model program (MMP) in higher dimensions.
1980s-1990s
Development of the minimal model program for threefolds (Kollár, Mori, Reid, Shokurov, etc.), culminating in the proof of existence of minimal models and abundance for threefolds.
2000s
Birkar, Cascini, Hacon, and McKernan prove the finite generation of the canonical ring for varieties of general type, a major breakthrough in the MMP for general type varieties.
2009
Hacon and McKernan prove the existence of flips for all dimensions, a key step in the MMP for arbitrary dimension.
2014
Birkar proves the boundedness of Fano varieties, leading to the proof of the Borisov-Alexeev-Borisov conjecture (BAB conjecture) for log Calabi-Yau varieties.
2016
Birkar, together with others, proves the existence of minimal models for varieties of general type in all dimensions (the so-called 'Birkar-Cascini-Hacon-McKernan' theorem extended).
2018
Birkar wins the Fields Medal for his contributions to birational geometry, including the proof of the BAB conjecture and advances in the minimal model program.
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