(The ruling is a toric morphism)
With the fan of \(\FF_a\) given by \(u_1 = (1,0)\), \(u_2 = (0,1)\), \(u_3 = (-1,a)\), \(u_4 = (0,-1)\), the lattice map \[ \phi : N = \ZZ^2 \to \ZZ, \qquad (p,q) \mapsto p \] sends \(u_1 \mapsto 1\), \(u_3 \mapsto -1\), and \(u_2, u_4 \mapsto 0\), so each cone of \(\FF_a\) lands in a cone of the fan \(\ts{\RR_{\geq 0}, \ts{0}, \RR_{\leq 0}}\) of \(\PP^1\). The induced toric morphism \(\pi : \FF_a \to \PP^1\) is the ruling, with invariant fibres \(D_1\) and \(D_3\) and disjoint invariant sections \(D_2\) and \(D_4\).
(The four charts)
Let \(\sigma_1 = \Cone(u_1, u_2)\), \(\sigma_2 = \Cone(u_4, u_1)\), \(\sigma_3 = \Cone(u_3, u_4)\) and \(\sigma_4 = \Cone(u_2, u_3)\). Then \[ U_{\sigma_1} = \Spec \CC[x, y], \quad U_{\sigma_2} = \Spec \CC[x, y^{-1}], \quad U_{\sigma_3} = \Spec \CC[x^{-1}, x^{-a}y^{-1}], \quad U_{\sigma_4} = \Spec \CC[x^{-1}, x^{a}y] , \] which patch as follows:
Setting \(y = 0\) gives the patching \(x \mapsto x^{-1}\) of a copy of \(\PP^1\), and patching in the fibre direction, for example \(U_{\sigma_1}\) with \(U_{\sigma_2}\), gives a copy of \(\CC \times \PP^1\); so \(\FF_a\) is a \(\PP^1\)-bundle over \(\PP^1\).
(The bundle)
\[ \FF_a \cong \PP\big( \OO_{\PP^1} \oplus \OO_{\PP^1}(a) \big) . \] The two disjoint sections are the ones cut out by the two summands, and they have \[ D_2^2 = -a, \qquad D_4^2 = +a , \] so \(D_2\) is the negative section. Its normal bundle is \[ N_{D_2 / \FF_a} \cong \OO_{D_2}(D_2) \cong \OO_{\PP^1}(-a) , \] because the normal bundle of a divisor in a surface is its own restriction, of degree \(D_2^2 = -a\) on \(D_2 \cong \PP^1\). Likewise \(N_{D_4/\FF_a} \cong \OO_{\PP^1}(a)\).
(Conventions, and what is invariant)
Twisting changes the bundle and not the surface: \(\PP(E) \cong \PP(E \tensor L)\) for any line bundle \(L\), so \[ \PP\big( \OO \oplus \OO(a) \big) \cong \PP\big( \OO(-a) \oplus \OO \big) , \] and both notations name \(\FF_a\). What does not depend on the convention is the pair of self-intersections \(\ts{-a, +a}\) and the difference \(a\) between them.
For \(a \geq 1\) the negative section is the unique irreducible curve on \(\FF_a\) with negative self-intersection, which is why \(\FF_a \cong \FF_b\) forces \(a = b\): the integer \(a\) is recovered from the surface. For \(a = 0\) there is no negative curve and \(\FF_0 = \PP^1 \times \PP^1\) has two rulings instead of one. Contracting the negative section of \(\FF_a\) gives the cone over the rational normal curve of degree \(a\), which is how the affine singular cone and the surface classification are the same picture seen twice.
Dependencies
FE-TORHIRZThe Hirzebruch surface F_a from its fanusesPR-TORMORToric morphisms, completeness, and invariants read off the fanuses