83 problems

Let $k$ be a number field. A non-trivial isotrivial elliptic fibration is an elliptic fibration over $\mathbb{P}^1$ that is isotrivial but not trivial. Density conjecture. The $k$-…

Let $Z^{\mathrm{sym}}_{N,r}$ be the symmetric Hilbert modular surface, and consider each $(N,r)$ in parts (ii)–(iii) of the paper's classification theorem. A Jacobian elliptic fibr…

Multisection-index conjecture. One has

Let $X$ be a smooth elliptic Calabi–Yau $4$-fold, and consider its vertical four-cycles, meaning four-cycles arising from the vertical part of the elliptic fibration. Their interse…

Let $X^{(n)}$ be a smooth Calabi–Yau $n$-fold, and let $\hat{D}$ be a divisor on $X^{(n)}$. The divisor $\hat{D}$ is nef if it has nonnegative intersection with every curve on…

Let $f\colon X\to Z$ be a klt-trivial fibration of relative dimension $k$, with $(X,B)$ a klt projective pair whose boundary $B$ has rational coefficients. Let $r$ be the positive…

Let an elliptic fibration geometrically engineer a gauge group, and let its Mordell–Weil group be the group of rational sections of the fibration. The center of the gauge group is…

Let $X$ be a complex manifold elliptically fibered with at worst $I_{1}$ fibers over a normal analytic variety $B$ such that … Let $\alpha,\beta\in\mathop{\mathcal{S}h}_{an}(X)$ be…

Let $X$ be the weighted Fano threefold hypersurface considered in the paper, with numerical parameter $N$, and let $a_{2}$ and $a_{3}$ be the weights defining the weighted projecti…

Wilson's conjecture. The condition $h\cdot c_2(X)>0$ is not essential for $|Nh|$ to define an elliptic fibration

Let $X\to B$ be as in Theorem MMP, with $X$ a Calabi–Yau variety. Suppose that $B$ is birational to a Mori fiber space $B'\to U$ with fiber $F$, and let…

Let $\phi:Y\to B$ be an elliptic fibration and let $D\subset Y$ be a normal crossing divisor such that $\phi(D)$ is a divisor in $B$. Assume that the generic fiber of…

Let $B'$ be the base obtained by blowing up $B$ at five points, so that $h^{1,1}(B')=10$, and let the nonabelian gauge group be $SU(15)\times SU(10)^3\times SU(5)^3$. The nonabelia…

Let $Z$ be the compact surface serving as the base of an elliptic Calabi–Yau 3-fold, and let $\rho(Z)$ denote its Picard number. Morrison's maximal Picard rank conjecture. The maxi…

Let $\pi:X\to B$ be a Calabi–Yau fibration with relative dimension one and a section, and let $F_{g,\alpha}$ denote its $\pi$-relative Gromov–Witten potentials. Elliptic-fibration…

DHT conjecture. The surface $Y$ admits an elliptic fibration $\pi\colon Y\to\mathbb{P}^1$ and an allowable loop $\gamma\subset\mathbb{P}^1$ compatible with the $\check{L}$-quasi-po…

Mirror symmetry for lattice-polarised weak del Pezzo surfaces. The mirror to $(V,C)$ is given by an elliptic fibration $W\to\Delta$ with fibre $F_p$ such that $\phi$ is a quasi del…

Let $X$ be an elliptic K3, Enriques or Coble surface, and let $\operatorname{Aut}(X)$ be its automorphism group. The Mukai conjecture. The virtual cohomological dimension…

Let $R\to\mathbb P^1$ be a general rational elliptic surface, and let sections mean section curves of this elliptic fibration. Transversality conjecture. Any two sections meet tran…

Relative holomorphic anomaly conjecture. The series lies in

Boundedness–Reid philosophy conjecture. Boundedness for Calabi–Yau varieties and Reid's philosophy are related.

Charged-multiplet conjecture. The charged multiplets $H_{ch}$ in the massless spectrum satisfy

Let $(A_i,B_i)$, for $i=1,\ldots,12$, be one of the twelve Type $i$ KSS pairs of smooth elliptic Calabi--Yau $3$-folds over $\mathbb{P}^2$. The pair is called nonbirational when no…

Let $J_i\to\mathbb{P}^2$ be the common relative Jacobian of the Type $i$ KSS pair. The Tate--Shafarevich group is the group of torsors of the Jacobian fibration that are locally tr…

Let $(A_i,B_i)$, for $i=1,\ldots,12$, be one of the twelve Type $i$ KSS pairs of smooth elliptic Calabi--Yau $3$-folds over $\mathbb{P}^2$, with elliptic fibrations…