A divisor $D$ on a variety $X$ is said to be ample if a sufficiently high multiple of $D$ furnishes an embedding of $X$ into some projective space.
Let $D$ be the divisor on $\mathbb{C}^2$ given by the cuspidal cubic plane curve $D= \{ y^2 - x^3 =0 \}$. How does one check whether $D$ is (not) ample?
The standard way to answer this is to compactify to $\mathbb P^2$ and note that (the closure of) $D$ is cut out by a section of $L = \mathcal O_{\mathbb{P}^2} (3)$. Now $L$ is obtained by cubing (under tensor product) the ample generator $\mathcal O_{\mathbb{P}^2} (1)$ and is in particular very ample, inducing the Veronese embedding $\mathbb P^2 \hookrightarrow \mathbb P^9$ given by $[x:y:z] \mapsto [x^3:x^2y:x^2z:xyz:xy^2:y^3:xz^2:y^2z:yz^2:z^3]$. Restricting this entire story to $\mathbb C^2 \cong (\mathbb P^2 \setminus \left\{z=0\right\}) \subset \mathbb P^2$ proves that your initial $D$ is very ample hence ample.
To be clear, this answer is so simple because (1) you started with an affine variety that sits naturally in a projective space, and (2) all hypersurfaces of fixed degree in a fixed projective space are linearly equivalent. So the word "cuspidal" and the particular equation are irrelevant; the only descriptor that matters in this problem is the word "cubic."