assume we have an irreducible (even toric in my case) variety $X$ with an ample line bundle $L$. Let d an integer such that $L^d$ is very ample and consider the projective embedding associated to $L^d$ in a $\mathbb{P}^N$. Assume I know how can compute $\chi(L^d)$ and $\chi(\mathcal{O}_X)$. Define $\deg L^d=\chi(L^d)-\chi(\mathcal{O}_X)$. What is the difference between $\deg L^d$ and the degree of $X$ inside $\mathbb{P}^N$? are they the same?
Thnks