I am struggling to understand exactly what a group extension is (Hartshorne is quite abstract for me). Studying vector bundles I encounter the following problem:
Let $X = P^1 ×P^1$ over the complex number be a quadric surface. We denote by $l$ and $m$ the standard basis of Pic$(X)=Z^2$. So, the canonical class $K_X =−2l−2m$ and $l^2=m^2=0$ and $lm=1$. Let $E$ be a rank 2 vector bundle on $X$ given by a non-trivial extension $e\neq 0$: $$ 0 \to O_X(l-3m) \to E \to O_X(3m) \to 0 $$ and
apparently this extension exists because $$ \text{Ext}^1(O_X(3m), O_X(l-3m)) \cong H^1(X,O_X(l-6m)) \cong \mathbb{C}^{10} $$ My question is is there a way to understand intuitively what information this group extension conveys?
From the short exact sequence I know that gives the information that $O_X(l-3m) \backsimeq \frac{E}{O_X(3m)}$ but I do not know what else is hidden there. How can I quickly compute $c_1(E)$ and $c_2(E)$? And how does this extension actually change $E$?
I will not address the question of intuition, since most of the time it comes from understanding the technicalities. I do not know how much you know about extensions, so feel free to ask.
For vector bundles $E,F$ on a scheme $X$, $\mathrm{Ext}^i(E,F)=H^i(X,E^*\otimes F)$. In your case, thus, if $L,M$ are line bundles on $X$, $\mathrm{Ext}^1(L,M)=H^1(L^{-1}\otimes M)$. The Chern classes can be computed by Whitney sum formula. For any extension, $0\to M\to E\to L\to 0$, you have $c_1(E)=c_1(L\otimes M)$ and $c_2(E)=c_1(L)\cdot c_1(M)$. In particular, the Chern classes of $E$ do not depend on the extension, while the extensions may be very different.