Divisors on $\mathcal{M}_{g,g+1}$ and the Minimal Resolution Conjecture for points on canonical curves

Details Divisors on $\mathcal{M}_{g,g+1}$ and the Minimal Resolution Conjecture for points on canonical curves Divisors on $\mathcal{M}_{g,g+1}$ and the Minimal Resolution Conjecture for points on canonical curves

2001-04-19

arxiv.org/abs/math/0104187v2

Ann. Sci. Ecole Norm. Sup. 36 (2003), no.4, 553-581

Mathematics

Algebraic Geometry

33 pages, 1 figure, final version with minor expository changes, to appear in Ann. Sci. ENS (2003)

Scientific paper

We formulate a conjecture on the behavior of the minimal free resolutions of sets of general points on arbitrary varieties embedded by complete linear series, in analogy with the well-known Minimal Resolution Conjecture for points in projective space. We then study in detail the case of curves. Our main result is that the Minimal Resolution Conjecture holds for general points on any canonical curve. On the other hand, for curves embedded with large degree, it always fails at a well-specified spot in the Betti diagram. The results are shown via techniques involving theta divisors associated to semistable vector bundles, difference varieties in Jacobians, and divisor class calculations in moduli spaces of curves with marked points.

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