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3. Mukai conjecture and related problems


    1. Add remark

      Problem 3.1.

      Mukai Conjecture: If $X$ is smooth Fano, then $$\operatorname{dim}X+\rho(X)-\operatorname{index}(X)\rho(X)\ge0,$$ where equality holds if and only if $X\simeq(\mathbb{P}^n)^m$.

      New Conjecture: $-K_X=\sum v_iM_i$, $M_i$ nef Cartier, $\sum v_i\le\operatorname{dim}+\rho$. Equality holds if and only if $X\simeq\prod\mathbb{P}^{n_i}.$

      Does New Conjecture implies Mukai Conjecture?


        • Add remark

          Problem 3.2.

          If every extremal contraction of a Fano is fibre type, is the nef cone simplicial?

              Cite this as: AimPL: Higher-dimensional log Calabi-Yau pairs, available at http://aimpl.org/higherdimlogcy.