4 problems

Segre's conjecture. If ${\mathcal{L}}_{2,d}(-\sum_{i=1}^hm_iP_i)$ is special, then its general member is non-reduced; equivalently, by Bertini's theorem, the system has a multiple…

Let $X$ be a K3 surface with $H^2=n\in2{\mathbb{Z}}$, and let ${\mathcal{L}}={\mathcal{L}}^n(d,m_1,\dots,m_h)$ be the system of curves in $|dH|$ through general points…

Speciality conjecture. Let $\mathcal{L}:=\mathcal{L}_{(d_1,d_2,d_3)}(m_1,\dots,m_r)$ be a linear system in standard form and let $D$ be a divisor in its strict transform. If…

The annihilation conjecture. The element $\pi^{r_1}$ annihilates the structure sheaf of