[Paper Review] Nodal Curves on K3 Surfaces

Xi Chen|arXiv (Cornell University)|Nov 22, 2016

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper proves that on a general polarized K3 surface $(X,L)$, every irreducible component of the Severi variety $V_{L,g}$ of genus $g$ curves in $|L|$ contains a component of $V_{L,g-1}$, thereby establishing inductively that general members of every component are nodal. The key method uses degeneration to a Bryan-Leung K3 surface and monodromy arguments on stable curves in the moduli space $\overline{\mathcal{M}}_g$, showing that the image of the family meets the boundary divisor, which implies non-completeness of the intersection with linear systems through $g-1$ points.

ABSTRACT

In this paper, we study the Severi variety $V_{L,g}$ of genus $g$ curves in $|L|$ on a general polarized K3 surface $(X,L)$. We show that the closure of every component of $V_{L,g}$ contains a component of $V_{L,g-1}$. As a consequence, we see that the general members of every component of $V_{L,g}$ are nodal.

Motivation & Objective

• To correct a flawed claim in prior work that all curves in $|L|$ of genus $g>0$ on a K3 surface are nodal.
• To establish that for a general polarized K3 surface $(X,L)$, the closure of every component of the Severi variety $V_{L,g}$ contains a component of $V_{L,g-1}$.
• To use this result to prove inductively that general members of every component of $V_{L,g}$ are nodal for all $0 \leq g \leq p_a(L)$.
• To support the conjecture on the irreducibility of the universal Severi variety $\mathcal{V}_{L,g}$ on K3 surfaces by verifying one of its key conditions.

Proposed method

• Uses degeneration to a Bryan-Leung K3 surface with Picard lattice $\begin{bmatrix}-2&1\\1&0\end{bmatrix}$ to analyze limit linear systems.
• Constructs a family of curves over a disk $\mathcal{X} \to \Delta$ with central fiber a union of rational and elliptic curves.
• Analyzes the moduli map $\rho: S \to \overline{\mathcal{M}}_g \times T$ and shows its image intersects the boundary $\Delta_0 \times T$.
• Applies dimension and irreducibility arguments to show that $\rho^{-1}(\Delta \times T) \cap S_t \neq \emptyset$ for $t \neq 0$, implying non-completeness of $W \cap \Lambda_\sigma$.
• Uses the fact that $\dim \rho(S) = 2$ and $\dim \rho(G) = 1$ for components $G$ of $\rho^{-1}(\Delta_0 \times T)$ to derive a contradiction if the image avoids the boundary.
• Relies on the fact that nodal fibers in $|F|$ lead to rational tails in the limit, forcing the moduli point to lie in $\Delta_0$.

Experimental results

Research questions

• RQ1 Does every component of the Severi variety $V_{L,g}$ on a general polarized K3 surface contain a component of $V_{L,g-1}$?
• RQ2 Are general members of every component of $V_{L,g}$ nodal for all $0 \leq g \leq p_a(L)$?
• RQ3 Can the degeneration method used for rational curves be extended to higher genus curves on K3 surfaces?
• RQ4 Does the monodromy action on the nodes of rational curves in $|L|$ act as the full symmetric group $\Sigma_p$?
• RQ5 Does the closure of every component of $\overline{V}_{L,g}$ contain a component of $V_{L,0}$, supporting the irreducibility of the universal Severi variety?

Key findings

• The closure of every irreducible component of $V_{L,g}$ on a general polarized K3 surface contains a component of $V_{L,g-1}$, as shown via degeneration to a Bryan-Leung K3 surface.
• This containment implies by induction that all general members of every component of $V_{L,g}$ are nodal for $0 \leq g \leq p_a(L)$, correcting a flawed earlier claim.
• The moduli map $\rho: S \to \overline{\mathcal{M}}_g \times T$ has image of dimension 2 and intersects the boundary divisor $\Delta_0 \times T$ in dimension 1, which is essential to the argument.
• The existence of a point $b \in S_0$ where the fiber $Y_b$ has a nodal rational tail forces $\rho(b) \in \Delta_0 \times T$, ensuring the image meets the boundary.
• The contradiction derived from assuming $\rho^{-1}(\Delta_0 \times T) \subset S_0$ and $\rho$ constant on components shows that $W \cap \Lambda_\sigma$ cannot be complete.
• The result supports the conjecture that the universal Severi variety $\mathcal{V}_{L,g}$ is irreducible, as it verifies one of the two required conditions for Harris-type irreducibility proofs.