Many things in algebraic geometry can be proved using a degeneration to combinatorial objects like hyperplane arrangements, monomial ideals or toric varieties.

For instance, de Fernex-Ein-Mustata proved an inequality involving certain invariants of a singularity (e.g., the Samuel multiplicity and log canonical threshold) by degenerating to a monomial ideal. For monomial ideals the inequality is a simple consequence of the arithmetic means geometric means inequality!

Another example: A smooth hypersurface has no automorphisms, as can be shown by degenerating to a union of hyperplanes (which is rigid for high degree!).

Many things in algebraic geometry can be proved using a degeneration to combinatorial objects like hyperplane arrangements, monomial ideals or toric varieties.

For instance, de Fernex-Ein-Mustata proved an inequality involving certain invariants of a singularity (e.g., the Samuel multiplicity and log canonical threshold) by degenerating to a monomial ideal. For monomial ideals the inequality is a simple consequence of the arithmetic means geometric means inequality!

Another example: A generic smooth hypersurface has no automorphisms, as can be shown by degenerating to a union of hyperplanes (which is rigid for high degree!).