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Compactified symplectic leaves in bundle moduli spaces
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DOI：10.1515/advgeom-2025-0034.png)
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Abstract
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Let E be a rank-2 vector bundle over an elliptic curve E, decomposable as a sum of line bundles of degrees d' > d >= 2, and L the determinant of E. The subspace L(E) subset of Pn-1 congruent to PExt(1)(L, O-E) consisting of classes of extensions with middle term isomorphic to E is one of the symplectic leaves of a remarkable Poisson structure on Pn-1 defined by Feigin-Odesskii and Polishchuk, and all symplectic leaves arise in this manner, as shown in earlier work that realizes L(E) as the base space of a principal Aut(E)-fibration. Here, we embed L(E) into a larger, projective base space L(E) of a principal Aut(E)-fibration whose total space parametrizes sections of E. The embedding realizes L(E)subset of (L) over bar (E) as a complement of an anticanonical divisor Y (one of the main results), and we give an explicit description of the normalization of Y as a projective-space bundle over a projective space. For d=2,(L) over bar (E) is one of the three Hirzebruch surfaces Sigma i, i = 0, 1, 2; we determine which occurs when and hence also the cases when L(E) is affine. Separately, we prove that for d < n/2 the singular locus of the secant slice Sec(d,z)(E) subset of Pn-1, the portion of the d(th) secant variety of E consisting of points lying on spans of d-tuples with sum z is an element of E, is precisely Sec(d-2). This strengthens the result that L(E) is smooth, appearing in prior joint work with R. Kanda and S. P. Smith.
Keywords:
Elliptic curve
projective space
vector bundle
pairing
Poisson structure
symplectic leaf
characteristic class
Chern class
Chow ring
multiplicative sequence
Hirzebruch surface
secant variety
smooth locus
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