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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.