Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 4 / 3

The identification with ${\mathbf{P}}^1$ means that $C$ is a smooth plane conic. An arbitrary singular or nonreduced conic over an algebraically closed field cannot be identified with the projective line, so this is a necessary qualification implicit in the requested description.

Choose an equation $\ell =0$ for $L\subset {\mathbf{P}}^3$ and a degree-two equation $q=0$ whose restriction to $L$ defines $C$. Its homogeneous ideal in ${\mathbf{P}}^3$ is $(\ell ,q)$. These equations form a regular sequence: $\ell$ is a non-zero-divisor in the polynomial ring, and modulo $\ell$ the ring is a polynomial ring in three variables, in which the nonzero quadratic $q$ is again a non-zero-divisor. The generator classes therefore form a basis of the conormal sheaf, with their grading shifts:One can verify the absence of relations from the two-generator Koszul complex: a relation between $\ell ,q$ is a multiple of $(q,-\ell )$, whose coefficients become zero modulo ${\mathcal{I}}_C$. This is the normal sheaf of a projective complete intersection calculation. Dualizing the conormal sheaf gives

These twists are restrictions from ${\mathbf{P}}^3$, not intrinsic degree-one twists on ${\mathbf{P}}^1$. A line in the plane cuts the degree-two curve in two points counted with multiplicity, so the degree of a line bundle ${\mathcal{O}}_C(1)$ is two. The Picard group of the projective line then givesEquivalently, the degree-two Veronese embedding $[s:t]\mapsto [s^2:st:t^2]$ pulls each linear coordinate back to a quadratic. Thus the normal bundle of a smooth plane conic isThe same splitting follows from the normal sequence for the two embeddings:Its outer terms are ${\mathcal{O}}_{{\mathbf{P}}^1}(4)$ and ${\mathcal{O}}_{{\mathbf{P}}^1}(2)$. The extension group class lies in ${\mathrm{Ext}}^1(\mathcal{O}(2),\mathcal{O}(4))=H^1({\mathbf{P}}^1,\mathcal{O}(2))=0$, by the preceding Čech cohomology calculation, so the sequence splits. Both methods distinguish the ambient twist from the intrinsic degree on the curve.