Some results on fake quadrics

TL;DR

The paper develops explicit divisor criteria for fake quadrics, distinguishing even and odd Néron–Severi lattices, to characterize when divisors are effective or ample and to describe the nef, movable, and effective cones. It proves that odd-type fake quadrics admit no negative curves and that all fake quadrics fiber over $\mathbb{P}^1$, while simultaneously ruling out holomorphic embeddings into $\mathbb{P}^4$ for both types. It also establishes a bounded cohomology property for curves on these surfaces and discusses Mori dream space implications, contributing to the broader understanding of fake quadrics and their birational geometry.

Abstract

In this paper, we give a criterion to assess the effectiveness and ampleness of divisors on a fake quadric surface $S$, and then we establish a relationship between the cones: \[\mathring{\Eff}(S)=\Amp(S)\subset \SAmp(S)=\Mov(S) \subset \Nef(S)=\Eff(S)=\overline{\Amp(S)}. \] In particular, we prove that any fake quadric of odd type does not contain a negative curve. This result is central to our manuscript. As applications, first we give that any fake quadric is a fibration over $\mathbb P^1;$ Subsequently, we show that no fake quadric can be embedded in $\mathbb P^4$; Finally, we prove that the fake quadric $S$ possesses the bounded cohomology property. This property is characterized by the existence of a positive constant $c_{S}$ such that $$h^1(\CO_S(C))\leq c_S h^0(\CO_S(C))$$ for any curve $C \subset S$.

Figures (2)

• Figure 1: Effective cones of $\tilde{S}$.
• Figure 2: Cones of a fake quadric of odd type

Theorems & Definitions (44)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Theorem 2.2
• proof