Exceptionality, Strongness, and Asymptotic Rarity for the Bondal–Thomsen Collection

Let X be a smooth projective toric Fano variety over an algebraically closed field of characteristic zero, and let $$\Theta _X$$ be the set of distinct Bondal–Thomsen line bundles appearing on either factor of the Hanlon–Hicks–Lazarev resolution of $$\mathcal {O}_\Delta $$ . We prove that $$\Theta _X$$ admits an exceptional ordering if and only if $$ {{\,\textrm{Ext}\,}}_X^q(L,M)=0 \qquad \text {for all }L,M\in \Theta _X\text { and all }q>0. $$ Since the diagonal resolution implies that $$\Theta _X$$ generates $$D^b{{\,\textrm{Coh}\,}}(X)$$ , either condition is equivalent to the existence of a full strong exceptional ordering of $$\Theta _X$$ . We also give an equivalent toric criterion: for every extremal primitive relation $$ p_1+\cdots +p_h=b_1y_1+\cdots +b_ky_k, $$ one must have $$\sum _j b_j\le 1$$ . This gives a dimension-independent explanation for a phenomenon previously observed computationally for all smooth projective toric Fano varieties of dimensions at most five. We then study how frequently this condition occurs as the dimension increases. If f(n) denotes the proportion of smooth projective toric Fano n-folds for which the Bondal–Thomsen collection admits a full strong exceptional ordering, then $$ \lim _{n\rightarrow \infty }f(n)=0. $$ More precisely, the number of such n-folds is at most exponential in n, whereas the total number of smooth projective toric Fano n-folds is at least $$ \exp \!\bigl (n\log n-2n\log \log n-O(n)\bigr ). $$