The Eisenbud-Huneke-Ulrich conjecture and bounds on minimal generators

We prove the Eisenbud–Ulrich conjecture on the powers of linearly presented ideals in characteristic zero. More generally, we study stabilization bounds for powers of m-primary ideals in a polynomial ring with partially linear resolutions, making further progress toward the more general Eisenbud–Huneke–Ulrich conjecture. We prove a global generation result on relevant higher syzygy bundles, which is a crucial ingredient to the proof of the Eisenbud–Ulrich conjecture. We also establish lower bounds for the number of minimal generators in two settings: ideals generated by quadrics and ideals that are virtually linearly presented. Motivated by these results, we formulate a conjecture predicting sharp lower bounds for the number of minimal generators of such ideals.