Let ${\mathcal P} \subset {\mathbb R}^d$ be a lattice polytope of dimension $d$. Let $b({\mathcal P})$ denote the number of lattice points belonging to the boundary of ${\mathcal P}$ and $c({\mathcal P})$ that to the interior of ${\mathcal P}$. It follows from the lower bound theorem of Ehrhart polynomials that, when $c({\mathcal P}) > 0$, one has

\begin{eqnarray}

\label{formula}

{\rm vol}({\mathcal P}) \geq ((d-1) \cdot b({\mathcal P}) + d \cdot c({\mathcal P}) - d^2 + 2)/d!,

\end{eqnarray}

where ${\rm vol}({\mathcal P})$ is the (Lebesgue) volume of ${\mathcal P}$. Pick's formula guarantees that, when $d = 2$, the inequality ($\ref{formula}$) is an equality. One calls ${\mathcal P}$ Castelnuovo if $c({\mathcal P}) > 0$ and if the equal sign holds in ($\ref{formula}$). A quick introduction to Ehrhart theory of lattice polytopes will be presented. Furthermore, a historical background on polarized toric varieties, to explain the reason why one calls Castelnuovo, will be briefly reviewed.

No special knowledge will be required to understand the talk.