Hilbert series and Fano index
- Polarization and Fano index
- The primitive ample line bundle used here is $\mathcal{O}_X(1)$. The Fano index $r$ is the largest positive integer for which $\omega_X^\vee\cong\mathcal{O}_X(r)$. This does not require the Picard rank to be one.
- Hilbert series
- $$\begin{equation} \notag \operatorname{Hilb}_X(t)=\sum_{m\geq0}\mathrm{h}^0(X,\mathcal{O}_X(m))t^m =\frac{P_X(t)}{(1-t)^5}. \end{equation}$$ For $r>1$, these degrees differ from anticanonical degrees.
- Showing more terms
- The expansion initially shows degrees $0$ through $5$. Click the expansion to add four coefficients computed from the rational form; the rational expression itself is static.