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Wall crossing for moduli of stable log pairs. (English) Zbl 07734892

Compactifying moduli spaces is a central problem of algebraic geometry. It has long been apparent that moduli spaces often admit different compactifications depending on some choice of parameters, and so it is natural to ask how these compactifications and their universal families are related as one varies the parameters. The goal of the present article is to answer this question for compact moduli spaces of higher dimensional stable log pairs or stable pairs for short.
A stable pair is a pair \((X, \sum a_iD_i)\) consisting of a variety \(X\) and a \(\mathbb{Q}\)-divisor \(\sum a_iD_i\) satisfying certain singularity and stability conditions, which we will recall below. The standard example is a smooth normal crossings pair with \(0 < a_i \leq 1\) and \(K_X + \sum a_iD_i\) ample. Compact moduli spaces of stable pairs with fixed coefficient or weight vector \(\mathrm{a} = (a_1, \cdots, a_n)\) and fixed numerical invariants have been constructed using the tools of the minimal model program; see [Kol22] and Section 2. These moduli spaces are quite large and unwieldy in general, and so in practice one studies the closure of a family of interest inside the larger moduli space. Theorem 1.1 below summarizes our main results in a simplified, but typical situation. We will state our general results in Section 1.1.
Theorem 1.1. Let \((X, D_1,\cdots, D_n) \to B\) be a family of smooth normal crossings pairs over a smooth connected base \(B\), and let \(P\) be a finite, rational polytope of weight vectors \(\mathrm{a} = (a_1, \cdots, a_n)\) such that \(a_i < 1\) and \((X, \sum a_iD_i) \to B\) is a family of stable pairs for each \(\mathrm{a} \in P\). Let \(\mathcal{N}_\mathrm{a}\) denote the normalized closure of the image of \(B\) in the moduli space of \(\mathrm{a}\)-weighted stable pairs with universal family of stable pairs \((\mathcal{X}_\mathrm{a}, \sum a_i\mathcal{D}_i) \to \mathcal{N}_\mathrm{a}.\) Then there exists a finite, rational polyhedral wall-and-chamber decomposition of \(P\) such that the following hold:
(a) For \(\mathrm{a}, \mathrm{a}'\) contained in the same chamber, \(\mathcal{X}_\mathrm{a}\to\mathcal{N}_\mathrm{a}\) is canonical isomorphic to \(\mathcal{X}_\mathrm{a'}\to\mathcal{N}_\mathrm{a'}\).
(b) For \(\mathrm{a}, \mathrm{b} \in P\) contained in different chambers and satisfying \(b_i \leq a_i\) for all \(i\), there are canonical birational wall-crossing morphisms \[\rho_{\mathrm{b,a}} : \mathcal{N}_\mathrm{a} \to \mathcal{N}_\mathrm{b}\] such that for any third weight vector \(\mathrm{c}\) with \(c_i\leq b_i\), we have \(\rho_{\mathrm{c,b}}\circ\rho_{\mathrm{b,a}}=\rho_{\mathrm{c,a}}\). Moreover, the map \(\rho_{\mathrm{b,a}}\) is induced by a birational map \(h^{\mathrm{b,a}} :\mathcal{X}_\mathrm{a} \dashrightarrow \rho_{\mathrm{b,a}}^* \mathcal{X}_\mathrm{b}\) such that, for a generic \(u \in \mathcal{N}_\mathrm{a}\), the fiberwise map \(h^{\mathrm{b,a}}_u : (\mathcal{X}_\mathrm{a})_u \dashrightarrow(\mathcal{X}_\mathrm{b})_{\rho_{\mathrm{b,a}}(u)}\) is the canonical model of \(((\mathcal{X}_\mathrm{a})_u, \sum b_i(\mathcal{D}_i)_u)\).

MSC:

14J10 Families, moduli, classification: algebraic theory
14J17 Singularities of surfaces or higher-dimensional varieties
14E30 Minimal model program (Mori theory, extremal rays)
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