I'm working through the details of Deligne and Mumford's 69' paper, "The Irreducibility of the Space of Curves of Given Genus", and I had a few quick questions:

1) On p. 77, they claim that for $x$ a node, $\pi:C'\rightarrow C$ the blowing-up of $x$ and $x_1,x_2\in C'$ the two points of $\pi^{-1}(x)$. Then supposedly it's "easy" to show that for an invertible sheaf $\mathcal L$ on $C$, we have $$\rm{Hom}(\mathfrak m_x,\mathcal L)\cong H^0(C',\pi^*\mathcal L),\rm{Hom}(\mathfrak m_x^2,\mathcal L)\cong H^0(C',\pi^*\mathcal L(x_1+x_2)),$$ but I'm having trouble seeing it. I've tried to show it locally by assuming the curve looks like $xy=0$ in $\mathbb A^2$, but then I'm finding that on an affine open $U$ which trivializes $\mathcal L$ and doesn't contain $x$, then the restriction of such a homomorphism is equivalent to giving a section of $\mathcal L$ over $U$, and thus a section of $\pi^*\mathcal L$ over the isomorphic preimage of this open. But around $x$ I'm finding that such a homomorphism is equivalent to giving two functions, one on each branch, which vanish at $x_1,x_2$, which seems to be giving an isomorphism with $H^0(C',\pi^*\mathcal L(-x_1-x_2))$. I find the same thing for the square. Any help? I'm sure this is easy for experts...

On p. 78 2) At the beginning of \textit{Case 3} they claim, and I agree, that the restriction of $\omega_C^{-n}$ has degree at most -2 on each component $E$ of $C$. Shouldn't this automatically imply it has no sections on the preimage of that component in the blow-up? Why do they go on to speak about when $n=2$, the degree of $\omega_C$ is 1 on that component, and two poles are allowed? Furthermore, if the whole point is when $n\geq 3$, why is this necessary?

3) Later after the corollary on that page, why can we assume that the projective bundle $\mathbb P(\pi_*(\omega^n_{C/S}))$ is trivializable so we can actually get a morphism from $S$ to the Hilbert Scheme?

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