The Moduli Stack of Gieseker-sl2-bundles on a Nodal Curve Ii
نویسنده
چکیده
Let X0 be an irreducible projective nodal curve with only one singular point, and let P0 be a line bundle on X0. The moduli SUX0(r;P0) of rank r vector bundles on X0 with determinant P0 is not compact. In [A], using the technique of Kausz ([K1], [K2]), we constructed a compactification GSL2B(X0;P0) of SUX0(2;P0), and studied its structure. Surprisingly, despite its seemingly natural definition, GSL2B(X0;P0) is not a good compactification. It has two components and one of them is non-reduced. This means that if (X0,P0) is a degeneration of (Xb,Pb), where b ∈ B is a parameter and Xb (b 6= 0) is an irreducible smooth projective curve and Pb is a line bundle on Xb, GSL2B(X0;P0) is not a semistable degeneration of SUXb(2;Pb). In this paper, we introduce a new compactification of SUX0(2;P0), and prove that it gives a semistable reduction of the above degeneration. Moreover we prove a decomposition theorem for the generalized theta divisors on this new moduli space. The contents of the sections are as follows. In section 2, we introduce basic definitions. In section 3, we introduce the new compactification of the moduli of vector bundles. In section 4, we study the local structure of the moduli space. In section 5, we study the global structure of the moduli space. The arguments in section 4 and 5 are quite similar to those in [A]. That is why we omitted some details. In section 6, we prove a decomposition theorem for the generalized theta divisors. In section 7, we collected some facts about the compactification KSL2 of SL2 that are used in the preceding sections.
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