Extremal Betti numbers of symbolic powers of two-dimensional squarefree monomial ideals

Let [Formula: see text] be a two-dimensional squarefree monomial ideal in a polynomial ring [Formula: see text], where [Formula: see text] is a field. In this paper, we give explicit formulas for the extremal Betti numbers of the [Formula: see text]th symbolic power of [Formula: see text] for all [Formula: see text]. As a consequence, we characterize the rings [Formula: see text] which are pseudo-Gorenstein as sense of Ene et al. [Pseudo-Gorenstein and level Hibi rings, J. Algebra 431 (2015) 138–161]. We also provide a complete classification for the level property of the second symbolic power [Formula: see text]. In particular, we obtain a new algebraic-property of the unknown Moore graph of degree 57.