Some Obstructions to Solvable Points on Higher Genus Curves

It is known that for a curve defined over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℚ</mml:mi> </mml:math> of genus <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , there exists a point on the curve defined over a solvable extension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℚ</mml:mi> </mml:math> . We relate points on curves of genus <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math> over solvable extensions to the Bombieri–Lang conjecture. Specifically, we show that varieties parametrising points defined over extensions with a fixed solvable Galois group are of general type. Moreover, we show the existence of certain subvarieties in these varieties imply the existence of solvable morphisms from the curve.