Abstract
Fix integers d and g ≥ 2. Consider the stack \(\mathcal{J}_{d,g}\), called the universal Jacobian stack of genus g and degree d, whose section over a scheme S is the groupoid of families of smooth curves of genus g over S together with a line bundle of relative degree d.
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Notes
- 1.
In [Cap94], this variety is called the universal Picard variety and it is denoted by P d, g . We prefer to use the name universal Jacobian, and therefore the symbol J d, g , because the word Jacobian variety is used only for curves while the word Picard variety is used also for varieties of higher dimensions and therefore it is more ambiguous. Accordingly, we will denote Caporaso’s compactified universal Jacobian by \(\overline{J}_{d,g}\) instead of \(\overline{P}_{d,g}\) as in [Cap94] (see Fact 16.1).
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Bini, G., Felici, F., Melo, M., Viviani, F. (2014). Compactifications of the Universal Jacobian. In: Geometric Invariant Theory for Polarized Curves. Lecture Notes in Mathematics, vol 2122. Springer, Cham. https://doi.org/10.1007/978-3-319-11337-1_16
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