9 problems

Let $C$ be a curve of genus $3$, and let $C^{(2)}$ be its symmetric square. Denote by $\{K\}$ the paracanonical system, namely the system of divisors algebraically equivalent to a…

Lazarsfeld–Popa's conjecture. Under these assumptions,

Let $X$ be a smooth projective variety, let $K_X$ be nef and effective, and define the irregularity by $q(X)=h^1(\mathcal{O}_X)$. Refined indecomposability conjecture. If…

Let $X$ be a smooth complex projective variety of maximal Albanese dimension and general type, and suppose that $X$ is primitive, that $q(X)=\operatorname{dim}X$, and that its Alba…

Let $X$ be a minimal surface with irregularity $q>2$, and write $\Omega_X^i$ for the sheaf of holomorphic $i$-forms on $X$. Assume that … Here $S^2(C)$ denotes the second symmetric…

Let $X$ be a minimal generalized Lagrangian surface of general type. Thus, for a nonzero $w\in\ker\psi_2\subseteq H^0(X,\Omega_X^2)$ of rank $2$, let $V\subset H^0(X,\Omega_X^1)$ b…

Primitive-special and non-primitive conjecture. If $X$ is primitive and special, then

Numerical inequality conjecture. If

Mendes-Lopez–Pardini conjecture. There should be no examples with $q(X)\geq 4$ and with no irrational pencils of genus at least $2$.