Augmented and restricted base loci of cycles (Q2152476)

This paper is concerned with the study of algebraic cycles on higher-dimensional algebraic varieties, by introducing and studying certain sets called \textit{base loci}. Such base loci are subsets \(B \subset X\) attached to \(k\)-cycles on a projective variety \(X\) over an algebraically closed field, where \(1 \leq k \leq \dim(X)-1\). They are generalizations of previously defined notions of stable, augmented and restricted base loci of Cartier divisors \(D\) on \(X\). Indeed, restricted and augmented basi loci of Cartier divisors were introduced in Section 1 of the paper [\textit{L. Ein} et al., Ann. Inst. Fourier 56, No. 6, 1701--1734 (2006; Zbl 1127.14010)]. The goal of the author is to extend these definitions to arbitrary \(k\)-cycles, and to prove a number of interesting results about them. The study of ample line bundles has a long history. Similarly, nef, base-point-free, big and semi-ample line bundles are important objects that arise in many areas of algebraic geometry. To ask whether a line bundle satisfies one of these properties is to ask about its \textit{positivity}. The cone of divisors defined by such and other positivity conditions, on a smooth projective variety \(X\), have been the subject of a great deal of work. The same holds for cones of curves defined by similar positivity conditions. The study of positivity for cycles of higher codimension and dimension have started to come into focus only recently. One direction of such a generalization is the notion of `ampleness' of subschemes \(Y \subset X\) of a projective variety \(X\), see [\textit{R. Hartshorne}, Ample subvarieties of algebraic varieties. Notes written in collaboration with C. Musili. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0208.48901)] and [\textit{J. C. Ottem}, Adv. Math. 229, No. 5, 2868--2887 (2012; Zbl 1285.14006)]. This paper takes another direction, focusing on positivity properties of \(k\)-cycles \(\alpha = \sum_{i = 1}^m n_i [Z_i]\) on a smooth projective variety \(X\) over an algebraically closed field, and their relation with certain base loci \(B \subset X\) defined by these cycles. Recall that the \textit{augmented base locus} \(\textbf B_+(D)\) and the \textit{restricted base locus} \(\textbf B_-(D)\) of a \(\mathbb Q\)-Cartier \(\mathbb Q\)-divisor \(D\) on a normal complex projective variety \(X\) were defined and studied in [\textit{L. Ein} et al., Ann. Inst. Fourier 56, No. 6, 1701--1734 (2006; Zbl 1127.14010)]. For every \(\mathbb Q\)-divisor, one has \(\textbf B_-(D) \subset \textbf B+(D)\), and \(D\) is called \textit{stable} if equality holds (see [loc. cit.]). The first goal of this paper is to generalize these definitions to \(k\)-cycles on a projective variety of dimension \(n\) over any algebraically closed field, for arbitrary \(k\) with \(1 \leq k \leq n-1\). To explain this, let \(\mathcal Z_k(X)_{\mathbb R}\) be the vector space of real \(k\)-cycles, and let \(N_k(X)\) be its quotient by the numerical equivalence relation. For \(\alpha \in \mathcal Z_k(X)_{\mathbb R}\), let \([\alpha] \in N_k(X)\) be the numerical equivalence class of \(\alpha\). The key notion of this paper is: Definition 1.1. Let \(\alpha \in N_k(X)\). Set \(|\alpha|_{\mathrm{num}} = \{e \in \mathcal Z_k(X)_{\mathbb R} \colon e \text{ is effective and } [e] = \alpha\}\). The \textit{numerical stable base locus} \(\textbf B_{\mathrm{num}}(\alpha) \subset X\) of \(\alpha\) is defined as follows. We take \(\textbf B_{\mathrm{num}}(\alpha) = X\) if \(|\alpha|_{\mathrm{num}} = \emptyset\) and \(\textbf B_{\mathrm{num}}(\alpha) = \cap_e \mathrm{Supp}(e)\) otherwise, where \(e\) ranges over the elements in \(|\alpha|_{\mathrm{num}}\). The \textit{augmented} and \textit{restricted base loci} of \(\alpha\) are the respective loci \[ \textbf B_+(\alpha) = \bigcap_{A_1, \dotsc, A_{n-k}} \textbf B_{\mathrm{num}}(\alpha - [A_1 \cdots A_{n-k}])\text{ and } \textbf B_-(\alpha) = \bigcap_{A_1, \dotsc, A_{n-k}} \textbf B_{\mathrm{num}}(\alpha + [A_1 \cdots A_{n-k}]), \] where \(A_1, \dotsc, A_{n-k}\) run through all ample \(\mathbb R\)-Cartier \(\mathbb R\)-divisors on \(X\). This generalizes the definitions \(\textbf B_+(D)\) and \(\textbf B_-(D)\) for divisors \(D\) on \(X\) introduced in [\textit{L. Ein} et al., Ann. Inst. Fourier 56, No. 6, 1701--1734 (2006; Zbl 1127.14010)]. In Sections 3 and 4, Lopez proves a number of properties of these base loci, analogues of similar properties for divisors (see [loc. cit.]). The second goal of the author is to answer the following question: What are the positivity properties of \(\alpha \in N_k(X)\) when \(\textbf B_+(\alpha)\) or \(\textbf B_-(\alpha)\) are empty or properly contained in \(X\)? He proves (Theorem 1) that \[ P_k(X) = \{[A_1 \cdots A_{n-k}] + \beta \in N_k(X) \mid A_1, \dotsc, A_k \text{ ample }\mathbb R\text{-Cartier }\mathbb R\text{-divisors}, \beta \in N_k(X) \colon \textbf B_{\mathrm{num}}(\beta) = \emptyset \} \] is a convex cone in \(N_k(X)\), open and full-dimensional if \(X\) is smooth. In addition, the author proves that, for \(\alpha \in N_k(X)\), the following assertions are true (see Theorems 2 and 3): \begin{itemize} \item[1.] \(\textbf B_+(\alpha) \subsetneq X\) if and only if \(\alpha\) is big; \item[2.] \(\textbf B_+(\alpha) = \emptyset\) if and only if \(\alpha \in P_k(X)\); \item[3.] \(\textbf B_-(\alpha) \subsetneq X\) implies that \(\alpha\) is pseudo-effective; the converse is true if the base field is uncountable; \item[4.] \(\textbf B_-(\alpha) = \emptyset\) implies that \(\alpha \in \overline{P_k(X)}\), and the converse is true if \(X\) is smooth. \end{itemize} Finally, in analogy with the divisor case, the author defines a cycle \(\alpha \in N_k(X)\) to be \textit{stable} if \(\textbf B_-(\alpha) = \textbf B_+(\alpha)\) (see Definition 10.1), and proves several results on stable cycles generalizing some results of [loc. cit.]