Stability of nets of quadrics in ${\mathbb{P}}^5$ and associated discriminants

Abstract

Let $S$ be a complete intersection surface defined by a net $\Lambda$ of quadrics in ${\mathbb{P}}^5$. In this paper we analyze GIT stability of nets of quadrics in ${\mathbb{P}}^5$ up to projective equivalence, and discuss some connections between a net of quadrics and the associated discriminant sextic curve. In particular, we prove that if $S$ is normal and the discriminant $\Delta (S)$ of $S$ is stable then $\Lambda$ is stable. And we prove that if $S$ has the reduced discriminant and $\Delta (S)$ is stable then $\Lambda$ is stable. Moreover, we prove that if $S$ has simple singularities then $\Delta (S)$ has simple singularities.

27 pages. We add a new proposition (Proposition 3.2), and delete some overlap part with arXiv:1302.4804. arXiv admin note: substantial text overlap with arXiv:1302.4804 by other authors

References in corpus (2)

• Compact Moduli Spaces of Del Pezzo Surfaces and Kähler-Einstein metrics
• Finite Hilbert stability of (bi)canonical curves