Reference: Etale And Crystalline Companions, II
Let \(C\) be a proper curve of genus \(g\) over an arbitrary field \(L\) (of arbitrary characteristic).
The degree \(\deg(V)\) of a nonzero bundle \(V\) on \(C\) is defined as
\[ \deg V = \deg(\wedge^{\text{top}} V). \]
The slope \(\mu(V)\) of a nonzero bundle \(V\) on \(C\) is defined as
\[ \mu(V) = \frac{\deg V}{\text{rank}(V)}. \]
\(E\) is semistable if \(\mu(E)\ge \mu(E')\) for all nonzero subbundles \(E'\) of \(E\), stable if \(\mu(E)>\mu(E')\) for all nonzero proper subbundles \(E'\) of \(E\).
If \(F\) is a subbundle of \(E\) of the same rank, then \(\deg(F)\le \deg(E)\).
Side note: How can there be a subbundle of the same rank? I’m not sure what this means. Because \(\mathcal{O}(-1)\) is not a subbundle of \(\mathcal{O}\), since the quotient \(\mathcal{O}_P\) is not locally free and hence not a bundle.
If \(0\to F\to E\to G\to 0\) is a short exact sequence of bundles, then
\[ \deg(E) = \deg(F) + \deg(G), \quad \mathrm{rank}(E) = \mathrm{rank}(F) + \mathrm{rank}(G). \]
you can think of \(\mu(E)\) as an intermediate slope between \(\mu(F)\) and \(\mu(G)\). As a result of this viewpoint, we have
\(\mu(F)<\mu(E) \Leftrightarrow\mu(G)>\mu(E)\)
\(\mu(F)>\mu(E) \Leftrightarrow\mu(G)<\mu(E)\)
\(\mu(F)=\mu(E) \Leftrightarrow\mu(G)=\mu(E) \Leftrightarrow\mu(F)=\mu(G)\)
\(\deg(V^*) = -\deg(V)\) and \(\mu(V^*) = -\mu(V)\).
The HN polygon of \(E\) is the convex hull of the set of points \((\mathrm{rank}(E'), \deg(E'))\) where \(E'\) goes over all subbundles of \(E\).
What does this mean? By the formalism of stability in abelian categories, we can see that semi-stability implies that for any two term filtration \(0\to E'\to E\to E''\to 0\) we have \(\mu(E')\le \mu(E)\le \mu(E'')\). Think \(E'\) and \(E''\) be part of the slopes in the diagram, with \(E'\) connecting the origin and on the left of \(E''\), then this is similar to the convex polygon we are familiar with in Newton polygons. This clearly generalises to the case of many-term filtrations.
Using Riemann-Roch, we can prove certain existence. If \(E\) is a line bundle, a classical example of Riemann-Roch is that \(h^0(E)\ge \deg E + 1 - g\), so \(\deg E\ge g\Rightarrow H^0(E)\ne 0\).
Let \(E\) be a bundle on \(C\), if all slopes of \(\mathrm{HN}(E)>2g-1\), then \(E\) is generated by global sections (that means there exists a surjection \(\mathcal{O}^I\to E\) for some index set \(I\)).