The Minimal Model Program is the core algorithm of birational geometry — the systematic procedure for simplifying algebraic varieties by contracting curves and performing flips until a minimal model or Mori fiber space is reached. In dimension three, the MMP has been a theorem since the late 1980s. But it has never been an algorithm. The proofs are existential: they show flips exist without giving a procedure to find them.
Cascini, Mayeux, and Sheridan make it effective. They give the first computational procedure for running the MMP on threefolds defined over the algebraic numbers. The algorithm takes a projective threefold with terminal singularities and explicitly computes the sequence of divisorial contractions, flips, and fiber space structures that constitute the MMP output.
The main technical obstacle is computing whether a flip exists and constructing it. The classical proof of flip existence in dimension three uses subtle vanishing theorems and deformation arguments that are not constructive. The algorithm replaces these with explicit algebraic computations: finding the flipping curve, computing the relevant graded rings, and verifying the flip termination conditions at each step.
Over the algebraic numbers, all the necessary arithmetic — computing intersections, checking ampleness, performing flips — can be done exactly. The algorithm outputs not just the final model but the complete MMP sequence, making the entire theoretical machinery into executable mathematics.
Forty years of theory. Now a program. The gap between “exists by theorem” and “computable by algorithm” was larger than it looked — and closing it required new computational techniques, not just patience.