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Let $E$ be a rank 2, degree $d$ vector bundle over a genus $g$ curve $C$. Theloci of stable pairs on $E$ in class $2[C]$ fixed by the scaling action areexpressed as products of $\Quot$ schemes. Using virtual localization, thestable pairs invariants of $E$ are related to the virtual intersection theoryof $\Quot E$. The latter theory is extensively discussed for an $E$ ofarbitrary rank; the tautological ring of $\Quot E$ is defined and is computedon 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 $2^g$ line subbundles of maximal degree. Doubling the zerosection 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 stablepairs/Donaldson-Thomas theory on the total space of $E$. This endows theresidue invariants of $E$ with enumerative significance: they actually\emph{count} curves in $E$.

Maximal subbundles, quot schemes, and curve counting