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6. Modules over the Cox ring


    1. Add remark

      Problem 6.1.

      Given $M$ an $S$-module, $S=Cox(X)$,
      1. can we always find another $S$-module $M'$ satisfying \begin{itemize}
      2. $\bullet$ $\tilde{M}=\tilde{M}'$
      3. $\bullet$ $pdim(M')\geq dim(X)$ \end{itemize} This is known to be true if $X=\mathbb{P}^{n_1}\times ... \times \mathbb{P}^{n_r}$.

      4. What is the relation of this to examples of [Chardin-D’Cruz] where adding embedded components drops regularity?

      5. Potential mechanism by truncating $M$ to $(M)_{\geq d}$? (This works for $\mathbb{P}^{n_1}\times ... \times \mathbb{P}^{n_r}$’s)
      6. Alternatively, could there be an invariant of $\tilde{M}$ which obstructs existence of such an $M'$?

          Cite this as: AimPL: Syzygies and mirror symmetry, available at http://aimpl.org/syzygyms.