For all integers k > 0 k>0 , we prove that the hypergeometric function \[ I ^ k ( α ) = ∑ j = 0 ∞ ( ( 8 k + 4 ) j ) ! j ! ( 2 j ) ! ( ( 2 k + 1 ) j ) ! 2 ( ( 4 k + 1 ) j ) !   α j \widehat {I}_k(\alpha )=\sum _{j=0}^\infty \frac {\bigl ((8k+4)j\bigr )!j!}{(2j)!\bigl ((2k+1)j\bigr )!^2 \bigl ((4k+1)j\bigr )!} \ \alpha ^j \] is a period of a pencil of curves of genus 3 k + 1 3k+1 . We prove that the function I ^ k \widehat {I}_k is a generating function of Gromov–Witten invariants of the family of anticanonical del Pezzo hypersurfaces X = X 8 k + 4 ⊂ P ( 2 , 2 k + 1 , 2 k + 1 , 4 k + 1 ) X=X_{8k+4} \subset \mathbb {P}(2,2k+1,2k+1,4k+1) . Thus, the pencil is a Landau–Ginzburg mirror of the family. The surfaces X X were first constructed by Johnson and Kollár. The feature of these surfaces that makes our mirror construction especially interesting is that | − K X | = | O X ( 1 ) | = ∅ |-K_X|=|\mathcal {O}_X (1)|=\varnothing . This means that there is no way to form a Calabi–Yau pair ( X , D ) (X,D) out of X X and hence there is no known mirror construction for X X other than the one given here. We also discuss the connection between our construction and work of Beukers, Cohen and Mellit on hypergeometric functions.

For all k > 0 integer, we show explicitly that the hypergeometric functionis a period of a pencil of curves of genus 3k + 1. The function I k is the regularised I-function of the family of anticanonical del Pezzo hypersurfaces X = X 8k+4 ⊂ P(2, 2k + 1, 2k + 1, 4k + 1) and the pencil we construct is a candidate LG mirror of the elements of the family. The surfaces X were first constructed by Johnson and Kollár [17]. The main feature of these surfaces, which makes the mirror construction especially interesting, is that | − K X | = |O X (1)| = ∅; thus, there is no way to form a Calabi-Yau pair (X, D) out of X. We also discuss the connection between our constructions and the work of Beukers, Cohen and Mellit [1] on hypergeometric functions.

GIULIA GUGIATTI AND FRANCO ROTA A. Motivated by homological mirror symmetry, this paper constructs explicit full exceptional collections for the canonical stacks associated with the series of log del Pezzo surfaces constructed by Johnson and Kollár in [JK01]. ese surfaces have cyclic quotient, non-Gorenstein, singularities. e construction involves both the GL(2, C) McKay correspondence, and the study of the minimal resolutions of the surfaces, which are birational to degree two del Pezzo surfaces. We show that a degree two del Pezzo surface arises in this way if and only if it admits a generalized Eckardt point, and in the course of the paper we classify the blow-ups of P 2 giving rise to them. Our result on the adjoints of the functor of Ishii-Ueda [IU15] applies to any nite small subgroup of GL(2, C).