Deformations and moduli of irregular canonical covers with $$K^2=4p_g-8$$

Abstract In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying $$K^2 = 4p_g-8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>K</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>-</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> , for any even integer $$p_g\ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> . These surfaces also have unbounded irregularity q . We carry out our study by investigating the deformations of the canonical morphism $$\varphi :X\rightarrow {\mathbb {P}}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>φ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> </mml:math> , where $$\varphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified in Gallego and Purnaprajna (Trans Am Math Soc 360(10):5489-5507, 2008) into four distinct families, one of which is the easy case of a product of curves. The main objective of this article is to study the deformations of the other three, non trivial, unbounded families. We show that any deformation of $$\varphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of $$\varphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> is two-to-one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations of X are unobstructed even though $$H^2(T_X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> does not vanish. Consequently, X belongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality $$p_g &gt; 2q-4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> <mml:mi>q</mml:mi> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , with an irreducible component that has a proper "quadruple" sublocus where the degree of the canonical morphism jumps up. These components are above the Castelnuovo line, but nonetheless parametrize surfaces with non birational canonical morphisms. The existence of jumping subloci is a contrast with the moduli of surfaces with $$K^2 = 2p_g- 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>K</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , studied by Horikawa. Irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus $$g\ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational. Finally, our study shows that there are infinitely many moduli spaces with an irreducible component whose general elements have non birational canonical morphism and another irreducible component whose general elements have birational canonical map.

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