ℤ/2ℤ-Equivariant smoothings of cusp singularities

Source abstract

Let ( p ∈ X ) (p\in X) be the germ of a cusp singularity and let ι \iota be an antisymplectic involution, that is an involution such that there exists a nowhere vanishing holomorphic 2-form Ω \Omega on X ∖ { p } X\setminus \{p\} for which ι ∗ ( Ω ) = − Ω \iota ^*(\Omega )=-\Omega . Assume also that the involution is fixed point free on X ∖ { p } X\setminus \{p\} . We prove that a sufficient condition for such a singularity equipped with an antisymplectic involution to be equivariantly smoothable is the existence of a Looijenga (or anticanonical) pair ( Y , D ) (Y,D) that admits an involution free on Y ∖ D Y\setminus D and that reverses the orientation of D D .