Configurations and Theirs Equations
In previous sections we have seen how the nilpotent elements of \(\boldsymbol{\mathcal{G}}_{\mbox{ $\Gamma $}}\) , given by the values of morphisms d ± [see (5.26) and (5.35)], give rise to very rich algebraic and geometric structures on J(X; L, d) (to be
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Configurations and Theirs Equations
In previous sections we have seen how the nilpotent elements of G , given by the values of morphisms d ˙ [see (5.26) and (5.35)], give rise to very rich algebraic and geometric structures on J.X I L; d / (to be more precise on the relative tangent sheaf T ). These are sl2 -structures in 6, 7 and the stratification in 9 respectively. However, it is not clear yet that these nilpotent elements are useful for elucidating the properties of configurations on X , as it is the case with central elements of C , the center of GQ (see, for example, Corollary 4.13, Theorem 4.16). This section addresses this question and it can be viewed as a concrete application of the theory developed so far. Namely, we show how to use sl2 -subalgebras associated to the nilpotent elements, given by the values of morphisms d ˙ , to write down equations defining configurations arising from geometric considerations. The way to produce these equations is somewhat evocative of the classical method of Petri, which gives explicit equations of hypersurfaces of degree 2 (quadrics) and 3 (cubics) through a canonical curve.1 We also give explicit equations for hypersurfaces (of all degrees) passing through a given configuration. The equations, in general, might be quite complicated and not very illuminating. What is essential and different in our approach is that the main ingredient in getting those equations is representation theoretic. Namely, we exploit the decomposition of the space of functions on a configuration into the irreducible sl2 -submodules under the action of sl2 -subalgebras, associated to the nilpotent elements of G defined by the values of d ˙ . This point of view on obtaining equations of projective embeddings, to our knowledge, is new and seems to be quite fruitful for gaining insight into projective properties of configurations of points on surfaces, geometry of curves on surfaces and surfaces themselves (see 10.4, 10.5, 9.6).
1
See e.g., Mumford’s survey, [Mu], and the references therein for more details.
I. Reider, Nonabelian Jacobian of Projective Surfaces, Lecture Notes in Mathematics 2072, DOI 10.1007/978-3-642-35662-9 10, © Springer-Verlag Berlin Heidelberg 2013
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10 Configurations and Theirs Equations
10.1 Geometric Set-Up In this subsection we recall a geometric context of our constructions. Let be a r M component in Cad m .L; d / and let .ŒZ; Œ˛/ be a point in J (see 2.7 for notation). This gives a short exact sequence of sheaves on X 0
OX
EŒ˛
IZ .L/
0
(10.1)
corresponding to the extension class ˛ 2 ExtZ1 D Ext 1 .IZ .L/; OX /, where IZ is the sheaf of ideals of Z on X . The sheaf EŒ˛ sitting in the middle of (10.1) is locally free of rank 2 with Chern classes c1 .EŒ˛ / D L and c2 .EŒ˛ / D d :
(10.2)
From (10.1) it follows that EŒ˛ comes with a distinguished global section which we call e. This is the image of 1 2 H 0 .OX / under the monomorphism in (10.1). The epimorphism in that sequence can be now identified with the exterior product with th
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