4 problems

Asymptotic semistability conjecture. There exists an integer $m_0\gg0$ such that, for every $m\ge m_0$, the syzygy bundle $S_{\mathcal{E}(m)}$ is semistable. This is proposed as a…

Let $X$ be a smooth projective variety, and let $A$ and $P$ be line bundles on $X$. Assume that $A$ is very ample, and set $L_d:=dA+P$ for every positive integer $d$. Ein–Lazarsfel…

Cohomological criterion for syzygy bundles. The two cohomology groups are isomorphic if and only if they are both zero; consequently, the syzygy bundles $F_i$ are Steiner if and on…

Let $X$ be a smooth projective variety of dimension $n$, let $A$ be the ample line bundle from the preceding setup, and let $M_d$ denote the syzygy bundle associated with the relev…