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A stability condition for line bundles on reducible varieties. (English) Zbl 1423.14249

The problem of compactifying the Picard scheme of a family of varieties has been arisen a lot of research, particularly in the case of family of curves. Dealing with singular varieties introduces extra difficulties. For instance, as showed by T. Oda and C. S. Seshadri [Trans. Am. Math. Soc. 253, 1–90 (1979; Zbl 0418.14019)], the compactified Picard scheme for a reducible curve is by no means unique.
To bypass these difficulties, in this paper, which extends the main result of the author’s thesis, a notion of stability condition for line bundles on reducible varieties is defined. This stability condition extends the notion of “balanced multidegree” proposed by L. Caporaso [J. Am. Math. Soc. 7, No. 3, 589–660 (1994; Zbl 0827.14014)].
With this notion of stability at hand, the author fixes the following setting: a flat and proper family of varieties \(\mathfrak{X}\rightarrow S\) with \(S\) a smooth curve, with fibers being connected and reduced (but possibly reducible) varieties and with the special fiber \(X\) having non-trivial canonical sheaf. Under these conditions, the author shows that any line bundle on the generic fiber of \(\mathfrak{X}\) can be extended to a semistable line bundle on the special fiber \(X\). Moreover, the number of possible extensions is bounded by \(2^{n-1}\), where \(n\) is the number of irreducible components of \(X\). In particular, under mild numerical conditions, this extension is unique.

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
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