The text provides excerpts from a mathematical talk concerning the study of singularities of pairs in both positive characteristic ($p$) and characteristic zero number theory. The speaker explains that traditional methods for studying singularities in characteristic zero, such as the use of resolution of singularities and vanishing theorems of cohomology, are unavailable in positive characteristics, necessitating the creation of a "bridge" to transport results between the two fields. A key object of study is the pair $(X, \mathcal{A}^e)$, consisting of a variety $X$ and a formal product of ideals $\mathcal{A}$ with real exponents $e$, and the speaker introduces several key invariants like the log discrepancy and the minimal log discrepancy (MLD). The main goal is to prove that certain crucial theorems regarding properties like discreteness and the ACC (Ascending Chain Condition) hold true in positive characteristics, mirroring established results in characteristic zero via a novel lifting theorem.
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- The subject of the talk is the singularities of pairs in characteristic $p$ and characteristic zero.
- The primary approach to studying singularities in characteristic zero relies on the existence of resolution of singularity.
- Other key properties available in characteristic zero include generic smoothness of morphism.
- Characteristic zero studies also utilize Vanishing theorems of cohomology.
- These crucial properties—resolution, generic smoothness, and vanishing theorems—are not available in positive characteristics.
- The speaker's project aims to construct a "bridge" to transport statements proven in characteristic zero to positive characteristics.
- The core object of study is a pair, denoted $(A, \mathcal{J}^e)$.
- In this pair, $A$ represents a variety over a field.
- $\mathcal{J}^e$ is a product of ideals, written formally as $\mathcal{J}_1^{e_1} \dots \mathcal{J}_r^{e_r}$.
- The exponents $e_i$ in the multi-ideal are required to be positive real numbers.
- An ideal raised to a real exponent is considered only a formal product, and not necessarily an ideal itself.
- The concept of the pair originated within the field of birational geometry.
- Initial singularity studies naturally focused only on the variety $X$.
- Birational geometers then began studying the pair consisting of the variety and a divisor, $(X, D)$.
- The pair concept is convenient for utilizing the induction of dimension frequently used in birational geometry.
- The pair $(X, D)$ is equivalent to the pair consisting of $X$ and the defining ideal of $D$.
- This generalized naturally to the pair $(X, \mathcal{A})$, where $\mathcal{A}$ is simply an ideal.
- Using integer exponents in the pair is natural, as the result remains an ideal.
- Rational exponents are generally acceptable because raising an ideal to an integer multiple corresponding to the denominator still yields an ideal.
- Real exponents appear naturally in birational geometry, such as in the BCHM (Cascini-Hacon-McKernan-Mustaţǎ) context, when considering the limit of rational exponents.
- The speaker’s viewpoint is justified by a "surprising theorem of Sommer".
- For the purpose of the talk, the object is simplified to the setting where the variety $A$ is assumed to be smooth.
- $A$ is a smooth variety defined over $k$, where $k$ is a field of arbitrary characteristic.
- $E$ is defined as a prime divisor over $A$ with center at zero.
- The existence of such a prime divisor $E$ implies the existence of a birational modification $\pi: \tilde{A} \to A$ where $E$ is an irreducible divisor on the normal variety $\tilde{A}$.
- $v_E$ denotes the discrete valuation corresponding to the prime divisor $E$.
- For an ideal $\mathcal{A}$, $v_E(\mathcal{A})$ is defined as the minimum value of $v_E(x)$ for elements $x$ in $\mathcal{A}$.
- $k_E$ is the coefficient of $E$ in the relative canonical divisor component and is a non-negative integer.
- The log discrepancy for the pair is defined as $a_E(A, \mathcal{J}^e) = k_E + 1 - \sum e_i v_E(\mathcal{A}_i)$.
- This log discrepancy value is a real number, even if $e_i$ are real numbers.
- The Minimal Log Discrepancy (MLD) at point 0 is defined as the infimum of $a_E(A, \mathcal{J}^e)$ over all prime divisors $E$ over $A$ with center at zero.
- MLD is constrained to either be greater than or equal to zero, or equal to minus infinity.
- If one log discrepancy becomes negative, the MLD automatically becomes $-\infty$.
- A larger MLD value implies a better or "milder" singularity, meaning it is closer to being non-singular.
- A prime divisor computing MLD does not always exist.
- In characteristic zero, a prime divisor computing MLD always exists, relying on the existence of appropriate resolution of singularities.
- A pair is defined as log canonical if its MLD is greater than or equal to zero.
- Log canonical singularity is considered "marginally acceptable" in birational geometry.
- Talachita's result, for characteristic zero, states that the set of log canonical multi-ideals is a discrete set.
- In characteristic zero, the Log Canonical Threshold (LCT) is known to be a rational number.
- The main application of the bridge is demonstrating that the discreteness result (Talachita's theorem) holds for positive characteristic when $A$ is a smooth point with a perfect residue field.