Abstract
<p> Following a method introduced by Thomas-Vasquez and developed by Grundman, we prove that many Hilbert modular threefolds of geometric genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are of general type, and that some are of nonnegative Kodaira dimension. The new ingredient is a detailed study of the geometry and combinatorics of totally positive integral elements <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a fractional ideal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a totally real number field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the property that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="trace x y greater-than min upper I trace y"> <mml:semantics> <mml:mrow> <mml:mi>tr</mml:mi> <mml:mo> </mml:mo> <mml:mi>x</mml:mi> <mml:mi>y</mml:mi> <mml:mo>></mml:mo> <mml:mo movablelimits="true" form="prefix">min</mml:mo> <mml:mi>I</mml:mi> <mml:mi>tr</mml:mi> <mml:mo> </mml:mo> <mml:mi>y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {tr} xy > \min I \operatorname {tr} y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y much-greater-than 0 element-of upper K"> <mml:semantics> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo> ≫ </mml:mo> <mml:mn>0</mml:mn> <mml:mo> ∈ </mml:mo> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">y \gg 0 \in K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . </p>