3 problems

Let $Q$ be a singular point of the discriminant $\Sigma$ and let $X_Q$ be the corresponding fiber in $X$. Let $\ell^a_Q$ be the class of an irreducible component of $X_Q$ in…

Let $X$ be a minimal projective $3$-fold of general type, and let $p_g(X)$ denote its geometric genus. Write $\varphi_{14,X}$ for the rational map defined by the complete linear sy…

Let $X_n$ be the varieties in the series of $G_K$-Mori fiber spaces over $\mathbb{P}^1$ whose general fibers are isomorphic to the degree-$2$ del Pezzo surface $S_K$. Nonrationalit…