Assume I have a planar projective curve $C\subseteq \mathbb{P}^2$. Furthermore, assume $C$ has a nodal singularity at some point $Q\in\mathbb{P}^2$. I am looking to resolve $C$'s nodal singularity by applying a blow up at the point $Q$, and my question is as follows.

Assume the blown up curve is $\tilde{C}$. I know that the following holds: $$0\longrightarrow\mathcal{O}_C\longrightarrow\pi_*\mathcal{O}_\tilde{C}\longrightarrow k(Q)\longrightarrow 0$$

Is it true that: $h_\tilde{C}(d) = h_C(d) + 1$?, where $h_\tilde{C}(d),h_C(d)$ are the corresponding Hilbert polynomials? My idea is that if this is true, it should follow from an exact sequence such as:

$$0\longrightarrow S(C)^d\longrightarrow \pi_*S(\tilde{C})^d\longrightarrow k\longrightarrow 0$$

However, it is not clear apriory why $\pi_*$ should be a degree 1, i.e. linear, map. I could imagine that this should follow from the fact that $\pi$ induces an isomorphism between $C$ and its blow up $\tilde{C}$.


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