Liftings of ideals in positive characteristic to those in characteristic zero:Surface case

1. Problem Statement

The paper addresses the fundamental challenge in algebraic geometry of comparing singularity invariants between fields of positive characteristic ($p > 0$) and characteristic zero ($0$). Specifically, it focuses on pairs $(A, \mathfrak{a}^{\mathbf{e}})$ consisting of:

• $A$: A smooth algebraic surface (dimension 2).
• $\mathfrak{a}^{\mathbf{e}}$: A "multi-ideal" (a formal product of ideals with real exponents).

Key invariants of interest include:

• Log Discrepancy ($a(E; A, \mathfrak{a}^{\mathbf{e}})$): Measures the severity of singularities.
• Minimal Log Discrepancy (mld): The infimum of log discrepancies.
• Log Canonical Threshold (lct): The threshold where a pair ceases to be log canonical.

The Core Difficulty: While many results in characteristic zero (e.g., discreteness of mlds, existence of computing divisors) are well-established, their counterparts in positive characteristic are often difficult to prove directly due to the lack of resolution of singularities in higher dimensions and the failure of certain vanishing theorems. The paper seeks to bridge this gap by constructing a "lifting" from characteristic $p$ to characteristic $0$ that preserves these invariants.

2. Methodology: The "Skeleton" Lifting

The author introduces a novel method called "skeleton lifting" to transfer geometric objects from a field $k$ (char $p$) to $\mathbb{C}$ (char $0$).

A. Skeletons and Compatibility

• Skeleton: For a finitely generated $k$-algebra $R$, a subring $S \subset R$ is a skeleton if $S$ is a finitely generated $\mathbb{Z}/(p)$-algebra such that $k[S] = R$.
• Compatible Skeletons: A pair of skeletons $(S, \tilde{S})$ where $S \subset R$ and $\tilde{S} \subset \tilde{R}$ (over $\mathbb{C}$) are compatible if $\tilde{S}$ is $\mathbb{Z}$-flat at $p$ and $S \cong \tilde{S} \otimes_{\mathbb{Z}} \mathbb{Z}/(p)$.
• Lifting: An object in characteristic $p$ (e.g., an ideal, a point, a blow-up center) is "lifted" to characteristic zero by finding a corresponding object in the compatible skeleton $\tilde{S}$ that reduces modulo $p$ to the original.

B. Inductive Construction of Blow-ups

The paper constructs a sequence of blow-ups in characteristic zero that mirrors a given sequence in characteristic $p$:

1. Setup: Start with a sequence of blow-ups at closed points $P_i$ in affine space $A^N_k$.
2. Induction: Assume a skeleton sequence of varieties $B_i$ and $\tilde{B}_i$ exists up to step $i-1$.
3. Preserving Incidence: The crucial step is lifting the center of the next blow-up, $P_i$, to a point $\tilde{P}_i$ in the characteristic zero skeleton such that the incidence relations are preserved (i.e., $P_i$ lies on the strict transform of an exceptional divisor $E_j$ if and only if $\tilde{P}_i$ lies on $\tilde{E}_j$).
4. Prime Lifting: Using properties of regular local rings and regular systems of parameters, the author proves that if $P_i$ is a regular point, a "prime lifting" of its defining ideal exists in the skeleton, ensuring the geometry of the blow-up is preserved.

C. Dimensional Restriction

The main theorem is proven specifically for dimension $N=2$ (surfaces). The author notes that for $N \ge 3$, a counterexample by Kollár exists where the valuation of a lifted polynomial does not match the original, suggesting the statement requires modification in higher dimensions.

3. Key Contributions and Theorems

Theorem 1.1 (Main Lifting Theorem)

For a sequence of blow-ups at closed points in $A^2_k$ (char $p$), there exists a corresponding sequence in $A^2_{\mathbb{C}}$ (char $0$) such that:

1. The varieties reduce modulo $p$ to the original ones.
2. The exceptional divisors $\tilde{E}_i$ are prime liftings of $E_i$.
3. Discrepancy Preservation: The log discrepancy coefficients are identical: $k_{\tilde{E}_i} = k_{E_i}$.
4. Valuation Preservation: For any polynomial $f \in k[x_1, x_2]$, there exists a lifting $\tilde{f} \in \mathbb{C}[x_1, x_2]$ such that the valuation along the exceptional divisors is preserved: $v_{E_i}(f) = v_{\tilde{E}_i}(\tilde{f})$.

Corollary 1.2 (Lifting of Multi-ideals)

Extends the lifting to multi-ideals. If $(A, \mathfrak{a}^{\mathbf{e}})$ is a pair in char $p$, there exists a pair $(\tilde{A}, \tilde{\mathfrak{a}}^{\mathbf{e}})$ in char $0$ such that:

• $v_E(\mathfrak{a}) = v_{\tilde{E}}(\tilde{\mathfrak{a}})$.
• $a(E; A, \mathfrak{a}^{\mathbf{e}}) = a(\tilde{E}; \tilde{A}, \tilde{\mathfrak{a}}^{\mathbf{e}})$.

4. Results and Applications

A. Discreteness of Log Discrepancies (Corollary 1.3)

• Result: The set of log discrepancies $\Lambda_{A, \mathbf{e}, a}$ for pairs on a smooth surface in positive characteristic is finite (and thus discrete).
• Significance: This provides a new proof for the discreteness of mlds in positive characteristic, a property previously known in char 0. It implies the existence of a prime divisor computing the mld.

B. Inclusion of Invariants (Corollary 1.4)

The paper establishes the following inclusions:

1. $\{ \text{mld}(0; A, \mathfrak{a}^{\mathbf{e}}) \mid \text{char } p \} \subseteq \{ \text{mld}(0; \tilde{A}, \tilde{\mathfrak{a}}^{\mathbf{e}}) \mid \text{char } 0 \}$
2. $\{ \text{lct}(0; A, \mathfrak{a}^{\mathbf{e}}) \mid \text{char } p \} \subseteq \{ \text{lct}(0; \tilde{A}, \tilde{\mathfrak{a}}^{\mathbf{e}}) \mid \text{char } 0 \}$

• Significance: This shows that the "bad" singularities (low mlds) or "sharp" thresholds (low lcts) found in positive characteristic are also present in characteristic zero. While the reverse inclusion was known via toric methods, this paper proves it via a geometric lifting method.

C. Campillo's Complex Model (Corollary 1.5)

• Result: Every plane curve singularity in positive characteristic admits a "complex model" (a curve in $\mathbb{C}^2$) with the same equisingular type (same multiplicity data in the resolution).
• Significance: This re-proves a result by Campillo using the skeleton lifting method, demonstrating the power of the technique to transfer equisingularity types.

5. Significance and Impact

• Bridging Characteristics: The paper provides a robust geometric mechanism to transfer problems from positive characteristic to characteristic zero, where powerful tools (like resolution of singularities and vanishing theorems) are available.
• New Proofs: It offers alternative, non-toric proofs for the discreteness of log discrepancies and the containment of mld/lct sets, expanding the toolkit available for studying singularities in positive characteristic.
• Limitations and Future Work: The author explicitly acknowledges that the method relies on the dimension being 2. The existence of Kollár's counterexample in dimension 3 suggests that a direct valuation-preserving lifting is impossible in higher dimensions. The paper sets the stage for a modified statement for $N \ge 3$ in future work.
• Technical Innovation: The concept of "skeletons" and "prime liftings" in the context of blow-up sequences is a sophisticated contribution to the theory of jet schemes and arc spaces, allowing for precise control over the reduction modulo $p$.

In summary, Ishii's work successfully establishes that for smooth surfaces, the singularity invariants in positive characteristic are "controlled" by those in characteristic zero, providing a rigorous framework for comparing these two distinct algebraic worlds.