# Blow up of a planar curve at a singularity

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 $$\tilde{C}$$.