Seminar
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| Location: | SLMath: Eisenbud Auditorium, Online/Virtual |
Vanishing Orders of Functions Along Singularities and Ideal Containment Problems in Noetherian Rings
In this talk, we delve into the intricate interplay between vanishing orders of functions along singularities and ideal containment problems in Noetherian rings. We start by focusing on the polynomial ring R = C[x1, . . . , xn] and explore the behavior of prime ideals and their symbolic powers. If P is a prime ideal then the nth symbolic power P(n) is the ideal of polynomial functions that vanish to order at least n at the generic point of V (P ). The Zariski-Nagata Theorem, established by Zariski in 1949, reflects the non- singular nature of n-space by asserting that if P ⊆ Q are prime ideals, then P (n) ⊆ Q(n) for all natural numbers n. However, when dealing with rings with singularities, it is no longer the case that P (n) ⊆ Q(n) for all prime ideal containments P ⊆ Q.
Pioneered by Huneke, Katz, and Validashti, The Uniform Chevalley Theorem introduces a linear adjust- ment, requiring functions to vanish along the generic point of V (P ) to ensure a vanishing order of at least n along the generic point of V (Q). Specifically, it establishes the containment of ideals as P(Cn) ⊆ Q(n) for all P ⊆ Q and n ∈ N, where C depends on the singular prime Q.
A notable limitation of the theorem is the dependency of the constant C on the singular prime Q, which restricts its applicability. This talk will explore methodologies to determine a constant C that remains inde- pendent of the prime Q, thereby enhancing the theorem’s utility. The investigation will also shed light on the role of the Izumi-Rees Theorem and unveil novel uniform properties exhibited by affine rings.
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