This defense explores the computation of the minimal exponent of a hypersurface singularity using birational geometry. Although the minimal exponent is originally defined through the Bernstein–Sato polynomial, we show that in several important cases it can be detected directly on a log resolution.

For isolated quasi-homogeneous singularities, we demonstrate that a single weighted blow-up produces an exceptional divisor that computes the minimal exponent. Building on this, we utilize the Mustață–Chen birational formula and Chen’s inversion of adjunction to formulate a squeeze criterion. Finally, we apply this criterion to ADE singularities and extend the results to certain Newton-degenerate cases, such as Cayley cubic singularities.