2011/03/10 by W. D. Gillam, Gillam, W. D.
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1103.2169
openalex publication_date 2011/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let E be a rank 2, degree d vector bundle over a genus g curve C. The loci of stable pairs on E in class 2[C] fixed by the scaling action are expressed as products of \Quot schemes. Using virtual localization, the stable pairs invariants of E are related to the virtual intersection theory of \Quot E. The latter theory is extensively discussed for an E of arbitrary rank; the tautological ring of \Quot E is defined and is computed on the locus parameterizing rank one subsheaves. In case E has rank 2, d and g have opposite parity, and E is sufficiently generic, it is known that E has exactly 2g line subbundles of maximal degree. Doubling the zero section along such a subbundle gives a curve in the total space of E in class 2[C]. We relate this count of maximal subbundles with stable pairs/Donaldson-Thomas theory on the total space of E. This endows the residue invariants of E with enumerative significance: they actually count curves in E.