M2 Normal Bundle Extra Light Mode Test

For two sheaves, choose injective resolutions $I_1^{\bullet }$ and $I_2^{\bullet }$ as in [F4]. Their termwise finite direct sum is injective: $\mathrm{Hom}(-,I_1^q\oplus I_2^q)$ is the finite direct sum of exact Hom functors. It resolves the direct sum sheaf. Global sections commute with finite direct sums, and kernels and images of the resulting pair of cochain maps are the direct sums of their kernels and images. Thus $H^q(C,F_1\oplus F_2)=H^q(C,F_1)\oplus H^q(C,F_2)$; applying this twice handles rank three. By [F2] with $A=\mathbb{C}$, $n=1$, and $d=0,-1,-2$, the three line summands have $(h^0,h^1)(\mathcal{O})=(1,0)$, $(h^0,h^1)(\mathcal{O}(-1))=(0,0)$, and $(h^0,h^1)(\mathcal{O}(-2))=(0,1)$. Hence $(h^0(N_1),h^1(N_1))=(1,0)$ and $(h^0(N_2),h^1(N_2))=(2,1)$. The ranks are three by the displayed direct sums, and additivity of degree gives $\mathrm{deg}{N}_1=0-1-1=-2=\mathrm{deg}{N}_2$.

On the chart with coordinate $z$, use the canonical frame $dz$; near infinity put $w=1\mathrm{/}z$, so $dz=-w^{-2}dw$. The canonical-line transition is therefore $e_{\mathrm{\infty }}=-z^{-2}{e}_0$. Multiplying one frame by the nonzero constant $-1$ gives the standard transition $z^{-2}$ from [F3], hence $K_C\cong \mathcal{O}(-2)$. Therefore $\det N_1\cong \mathcal{O}(-2)\cong K_C$ and $\det N_2\cong \mathcal{O}(-2)\cong K_C$.

On a trivializing affine chart $U=\mathrm{Spec}R$, put $S=R[v_1,v_2,v_3]$ and let $J\subset S\otimes A$ define a flat embedded deformation of the zero section over an Artinian local $\mathbb{C}$-algebra $A$ with maximal ideal $\mathfrak{m}$. Write $M=(S\otimes A)\mathrm{/}J$. The projection induces $f:R\otimes A\to M$ reducing to the identity on $R$. Flatness gives ${\mathfrak{m}}^{k}M\mathrm{/}{\mathfrak{m}}^{k+1}M\cong ({\mathfrak{m}}^k\mathrm{/}{\mathfrak{m}}^{k+1}){\otimes }_{\mathbb{C}}R$: tensor the inclusions of successive ideals with $M$ and then reduce modulo $\mathfrak{m}$. The map $f$ induces the identity on each such quotient. Since $\mathfrak{m}$ is nilpotent, finite induction on this filtration makes $f$ an isomorphism. The images of the $v_j$ are therefore unique $t_j\in \mathfrak{m}(R\otimes A)$, and $J=(v_1-t_1,v_2-t_2,v_3-t_3)$. Conversely this ideal has quotient $R\otimes A$, which is free over $A$ and hence flat. For the dual numbers, $t_j=\epsilon s_j$ with $s_j\in R$. On overlaps the vectors $s$ transform by the vector-bundle transition, so their compatible graphs are precisely global sections of $N_i$. Addition and scalar multiplication of these coefficients identify the first-order tangent space with $H^0(C,N_i)$.

In a local trivialization of a rank-three algebraic vector bundle $N$, the total-space transition has the form $(z,v)\mapsto (\varphi (z),G(z)v)$, where $G(z)\in G{L}_3(\mathbb{C})$. Its tangent transition is block triangular with diagonal blocks ${\varphi }^{\mathrm{\prime }}(z)$ and $G(z)$. Along the zero section the vertical tangent is $N$ and $d\pi \circ d{s}_0$ is the identity on $T_C$, so $s_0^{\ast }T\mathrm{Tot}(N)=d{s}_0(T_C)\oplus N$. The normal quotient defined in the Statement is therefore naturally $N$. The tangent quotient by the vertical bundle is ${\pi }^{\ast }{T}_C$, and the cotangent determinant has transition $K_{\mathrm{Tot}(N)}\cong {\pi }^{\ast }(K_C\otimes (\det N{)}^{\ast })$, where $\pi :\mathrm{Tot}(N)\to C$ is projection. Substituting $\det N_i\cong K_C$ from step 1.2 makes this line bundle trivial for each $i$.

Apply step 1.3 on both affine charts to the supplied lifts over $A^{\mathrm{\prime }}\twoheadrightarrow A$. Two lifts of the same graph differ by a vector in $R{\otimes }_{\mathbb{C}}I$. On overlaps this difference transforms by $N_i$; the condition ${\mathfrak{m}}_{A^{\mathrm{\prime }}}I=0$ makes it independent of the original nilpotent coefficients. Thus the pairwise differences form a Čech cocycle in $N_i\otimes I$. Changing local graphs by sections of $N_i\otimes I$ changes the cocycle by their coboundary. A compatible glued lift gives zero class; conversely a zero class supplies graph corrections that make the lifts agree and glue. Since $I$ is finite-dimensional and $N_i$ splits into twisting lines, the standard-cover comparison in [F2], combined with the finite-sum argument of step 1.1, identifies this Čech group with $H^1(C,N_i\otimes I)$. This is an obstruction space, not a claim that any nonzero obstruction occurs. For $I=\mathbb{C}$ it is $H^1(C,N_i)$; for $I=0$ both the difference class and group vanish. Step 1.1 gives the stated different dimensions. No physical spectrum follows.