Whenever one studies moduli spaces of vector bundles on curves, one of the first things to be introduced is the "slope" of a vector bundle, i.e., its degree:rank ratio. Is there a nice (preferably geometric) intuition behind the use of the word "slope" for this?
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This is an addition to Angelo's comment. Given a vector bundle $E$, you can consider its Harder-Narasimhan filtration and assign to each element of the filtration a point in the degree-rank plane. The HN-polygon is the polygon obtained by connecting the dots. S.S.Shatz discussed the behaviour of the HN-polygon under specialisation in The decomposition and specialisation of algebraic families of vector bundles I beleive this is where the term orginated, see also Atiyah-Bott, section 7 (p.565). Addendum: You can gain minor visual gratification from looking at the degree-rank plane as follows. If $F\subset E$ is a subbundle, then $$ \deg \underline{Hom}(F,E)= \textrm{rk}F\deg E-\textrm{rk}E \deg F= \left| \begin{array}{cc} \deg E & \deg F\\ \textrm{rk} E & \textrm{rk} F\\ \end{array} \right|. $$ Also, $F$ destabilises $E$ exactly when the above determinant has negative sign. |
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`slope'' is justified by the fact that it measures the relative growth of polyvectors (along orbits of the Kontsevich-Zorich cocycle) inside flat subbundles of the Hodge bundle over the moduli space of Abelian differentials with respect to their`counterparts'' inside the tautological subbundle. – Matheus Oct 3 at 17:11