23 problems

Grinenko's conjecture. $X$ is birationally rigid if and only if it satisfies the $K$-condition.

Let $\mathcal{F}$ be the projective bundle used to construct the threefold Mori dream space $X$, and let $f$ and $g$ be the defining equations of the two hypersurfaces whose comple…

Semistable dP3 rigidity conjecture. Then $X/\mathbb{P}^1$ is birationally rigid. This is a semistable variant of Grinenko's conjecture.

Let $U$ be the double space considered in the source, and let ${\mathcal S}$ denote the family of curves used there to construct the fibrations $V_l/\mathbb{P}^1$. Khashin's conjec…

Let $X$ be the Fano threefold under consideration, determined by the parameters $(a_1,a_2,a_3,b_1,b_2,b_3)$, and let $\eta\colon C\to\mathbb{P}^1$ be the associated morphism. Let…

Let $\pi:\mathcal{X}\to B$ be a smooth del Pezzo fibration. Fix an intersection profile $\lambda$, let $N_\lambda$ be the affine subset of $N_1(\mathcal{X})$ consisting of curve cl…

Let $d$ be an integer with $1\leq d\leq 3$. Let $X\to\mathbb P^1$ be a $\mathfrak D$-semistable del Pezzo fibration of degree $d$, and assume that it satisfies the $K$-condition, m…

Weak Approximation conjecture. Every possible intersection profile can be realized by some section.

Abel–Jacobi MRC conjecture. In this situation, the Abel–Jacobi map always agrees with the MRC fibration for a family of sections.

Dominant-families conjecture. There is some translate $\mathcal{T}$ of $\mathrm{Nef}_{\lambda}$ in $N_{\lambda}$ such that every numerical class in $\mathcal{T}_{\mathbb{Z}}$ is re…

Weak Approximation conjecture. Since the Brauer group of $k(\mathbb{P}^{1})$ is trivial, the condition that a section can intersect only components of fibers occurring with multipl…

Geometric Manin's counting conjecture. After removing the contributions of an exceptional set of curves, the asymptotic growth is expected to satisfy

Iskovskikh's conjecture. If

Let $X\to\mathbb{P}^1$ be a del Pezzo fibration of degree $1$ with at most terminal singularities. It satisfies the $K$-condition when $-K_X$ is not in the interior of the movable…

Let $X_n$ be the $\operatorname{PSL}_2(7)$-equivariant del Pezzo fibration of degree $2$ over $\mathds{P}^1$ defined by … where $a_{2n}$ is a homogeneous polynomial of degree $2n$…

A del Pezzo fibration is birationally rigid if it has only one Mori fiber space structure in its birational class. Let $X$ be a del Pezzo fibration satisfying the hypotheses of The…

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…

Let $G_K\cong \mathrm{PSL}_2(\mathbb{F}_7)$ be the Klein simple group, and let $X_n/\mathbb{P}^1$ be the $G_K$-Mori fiber spaces described above. A $G_K$-Mori fiber space is a thre…

Semistability-modified conjecture. If $X$ satisfies the $K$-condition, then $X$ is birationally rigid.

Irrationality conjecture. The varieties $\mathcal{X}_n$ are all irrational except for those two particular members; consequently, only two embeddings of…

Let $X$ be a smooth threefold Mori fibre space over $\mathbb{P}^1$, with fibres being smooth del Pezzo surfaces of degree $4$. Let $\operatorname{Mob}(X)$ denote the mobile cone of…

Let $X$ be a standard degree-three del Pezzo fibration, meaning a standard del Pezzo fibration whose generic fibre is a cubic surface. Bi-rigidity conjecture. The threefold $X$ is…

Let $f\colon X\to Z\ni o$ be the germ of a del Pezzo bundle, and let $m_oF_o=f^*(o)$ be a fiber of multiplicity $m_o$, where $m_o\geq 5$ and $F_o$ is of type $I_{2,3,6}$ or…