6 problems

Existence conjecture. There exists a stable rank-2 vector bundle $\mathcal{E}$ arising as the cohomology of a minimal monad of this form. Moreover, for all sufficiently large integ…

Let $(r,c,d)\in\mathbb{Z}\oplus\mathbb{Z}\oplus\tfrac{1}{2}\mathbb{Z}$, and suppose that there are Gieseker-semistable sheaves $E$ on $\mathbb{P}^3$ with … Write $E(r,c,d)$ for the…

Let $M(r,c,d,e)$ denote the moduli space of Gieseker-semistable sheaves on ${\mathbb{P}}^3$ with Chern character $(r,c,d,e)$. Let $r \in {\mathbb{Z}}_{\geq 0}$,…

Let $d$ be a positive integer, and let $W_{\mathbb{R}P^3}(d,l)$ denote the Welschinger invariant of degree $d$ with the parameter $l$ specifying the real-point constraint. Equality…

Let $c$ be the second Chern class, also called the charge, of a mathematical instanton bundle on $\mathbb{C}\mathbb{P}^{3}$. Let $\mathcal{M}_c$ denote the moduli space of mathemat…

Laface–Ugaglia conjecture. begin{enumerate} item If $q(D)\leq 0$, then $h^0(D)=h^0(D-Q)$. item If $q(D)>0$, then $D$ is special if and only if $d<m_1+m_2-1$, and in that case