22 problems

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…

Fix a degree $d$ and geometric genus $g$, and let the normalization of the Severi variety parametrize plane curves with these invariants. Let $A$, $B$, and $C$ be the divisor class…

Let $A$, $B$, and $C$ be the standard divisor classes on the Severi variety, and let the boundary divisors denote the divisors parametrizing reducible or otherwise degenerate curve…

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 $V(L)$ be the genus $0$ Severi variety of a line bundle $L$ on a smooth projective algebraic surface, and consider the Severi curves arising from rational bisections of ellipti…

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 ${\rm S}$ 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 ${\mathcal K}_p$ be the moduli space of polarized K3 surfaces $(X,L)$ of genus $p=p_a(L)$, and let … be the universal Severi variety of genus $g$ curves in $|L|$ over…

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 $X$ be a $K3$ surface or an Abelian surface, and let $\xi\in\operatorname{NS}(X)$. For $g>0$ in the $K3$ case, respectively $g>2$ in the Abelian case, consider the Severi varie…

Atiyah–Berndt conjecture. There should be a K-theoretic proof that

Let $Q\subset\mathbf{P}^3$ be a smooth quadric, let $B$ be a general curve of type $(4,4)$ on $Q$, and let $S_0\to Q$ be the double cover branched along $B$. Let $\mathfrak V$ be t…

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…

A family $f:S\to\Delta$ is called well behaved when, after suitable base changes and blow-ups, it admits a semistable model that gives a complete description of the limiting Severi…

Let $\Delta$ be a simplicial complex with $3a+3$ vertices, and let $L_\Delta^{\bullet}$ be its face complex, formed from the wedge products indexed by the faces of $\Delta$. Let…

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