01 · The idea
In everyday language
Different polynomial equations can describe varieties that agree away from lower-dimensional subsets. Birational geometry studies this shared structure instead of insisting that every point remain unchanged. The minimal model programme uses the canonical class to detect curves whose contraction can simplify a variety. Sometimes a direct contraction creates an obstruction; a flip replaces the local configuration with another birational model that lets the process continue. Mori’s existence theorem for three-dimensional flips made a crucial operation available. The outcome can be a minimal model or a Mori fibre space. Controlled singularities are part of the method, not an accidental failure of smoothness.
Why it matters
Mori’s methods turned the classification of threefolds into a systematic programme, providing operations and structural principles that became central to higher-dimensional algebraic geometry.
