6 problems

Let $S$ be a smooth projective complex surface with $K_S\not\equiv 0$ that admits no minimal elliptic fibration. Let $Z$ be the union of all $(-2)$-curves, and let…

K-equivalent autoequivalence-group conjecture.

Autoequivalence-group conjecture. There is a short exact sequence

The autoequivalence-group conjecture. If $N_S=2$, then

The exact-sequence conjecture. There should be a short exact sequence

Let $X$ be a smooth projective surface with $omega_X$ ample or anti-ample. For a positive integer $m$, let $X^{[m]}$ denote the Hilbert scheme of length-$m$ subschemes of $X$, and…