14 problems

Let $R$ be a rational elliptic surface, and let $V(R,L)$ denote the genus $0$ Severi variety associated to a line bundle $L$ on $R$. Reducibility conjecture. There exist line bundl…

Let $\mathcal{V}_{\bf r}$ be the Severi variety of unicuspidal rational curves of type $({\rm S},{\bf r})$, under the hypotheses and notation of the codimension conjecture above. U…

Let ${\bf r}=(r_1,\ldots,r_n)\in \mathbb{N}_{>0}^n$, and let $\mathcal{V}_{\bf r}\subset M^n_{d,g;\mathrm{S}}$ be the subvariety of maps $f:\mathbb{P}^1\rightarrow\mathbb{P}^n$ wit…

Let ${\rm S}$ be a numerical semigroup with conductor $c$, and let ${\bf k}=(k_1,\dots,k_n)$ be a ramification profile. Let $B$ be the set of Betti elements of the ramification ord…

Universal Severi irreducibility conjecture. For every $0\leq \delta\leq g$, the universal Severi variety $\mathcal{V}_{g,\delta}$ is irreducible.

The K3 Severi irreducibility conjecture. For every fixed $0\leq \delta\leq g-1$, the Severi variety $V_{\delta}(Y,L)$ is irreducible.

Let $X_0$ be a general hyperelliptic K3 surface, and let $Z_{2r}$ be the determinantal curve constructed to dominate the curve $\Gamma_{2r}$ parametrizing divisors in…

The ordered-node irreducibility conjecture. Then ${\mathcal W}_{L,0}$ is irreducible. This is identified as the difficult monodromy-related part of the proposed approach to the irr…

Let $A_{2k}$ be the indicated singularity, let $S$ be the base of its miniversal deformation, and let $D(\ell)\subset S$ denote the corresponding Severi stratum. For integers…

Let $S$ be the symplectic manifold with symplectic form $\Omega$, let $\chi$ denote the relevant matrix of logarithmic vector fields, and let…

Let $f:S\to\Delta$ be a semistable family of surfaces, endowed with a line bundle $\mathcal L$, and let $\delta$ be a positive integer. A semistable model is $\delta$-absolutely go…

Let $\mathcal{B}_g$ be the moduli stack of primitively polarized K3 surfaces of genus $g>2$, parametrizing pairs $(S,M)$ where $M$ is ample and primitive with $(M\cdot M)=2g-2$. Le…

Let $S$ be a toric surface over an algebraically closed field of characteristic zero, let ${\mathcal L}\in \operatorname{Pic}(S)$ be an effective class, and let $g\geq 0$ be a non-…

Asymptotic irreducibility conjecture. For every $\varepsilon>0$, if $k$ is sufficiently large with respect to $\varepsilon$, then for every integer $h$ satisfying