Some results on fake quadrics

TL;DR

This paper investigates the geometric properties of fake quadric surfaces, establishing relationships between various cones of divisors, proving the non-existence of negative curves in odd type cases, and exploring applications like fibrations, embeddings, and cohomology bounds.

Contribution

It provides new criteria for divisors on fake quadrics, characterizes their cone structures, and demonstrates several key geometric and cohomological properties.

Findings

Fake quadrics of odd type do not contain negative curves.

Any fake quadric admits a fibration over .

Fake quadrics cannot be embedded in .

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 $C \subset S$.