We give a complete classification of the spherical 3-manifolds that bound smooth rational homology 4-balls. Furthermore, we determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. To this end, we use constraints for 3-manifolds to bound rational homology balls induced from Donaldson’s diagonalization theorem and Heegaard Floer correction terms.

We give an explicit sequence of Heegaard moves interpolating between local versions of the Kauffman-states Heegaard diagram and the planar Heegaard diagram used in knot Floer homology, and show how these local moves can be used to translate between the global versions of the Heegaard diagrams.

In the paper [7] by Cook et al., which introduced the concept of unexpected plane curves, the focus was on understanding the geometry of the curves themselves. Here, we expand the definition to hypersurfaces of any dimension and, using constructions which appeal to algebra, geometry, representation theory, and computation, we obtain a coarse but complete classification of unexpected hypersurfaces. In particular, we determine each $(\mathit{n},\mathit{d},\mathit{m})$ for which there is some finite set of points $\mathit{Z}\subset {\mathbb{P}}^{\mathit{n}}$ with an unexpected hypersurface of degree d in ${\mathbb{P}}^{\mathit{n}}$ having a general point P of multiplicity m. Our constructions also give new insight into the interesting question of where to look for such Z. Recent work of Di Marca, Malara, and Oneto [10] and of Bauer, Malara, Szemberg, and Szpond [5] gives new results and examples in ${\mathbb{P}}^2$ and ${\mathbb{P}}^3$. We obtain our main results using a new construction of unexpected hypersurfaces involving cones. This method applies in ${\mathbb{P}}^{\mathit{n}}$ for $\mathit{n}\ge 3$ and gives a broad range of examples, which we link to certain failures of the Weak Lefschetz Property. We also give constructions using root systems, both in ${\mathbb{P}}^2$ and ${\mathbb{P}}^{\mathit{n}}$ for $\mathit{n}\ge 3$. Finally, we explain an observation of [5], showing that the unexpected curves of [7] are in some sense dual to their tangent cones at their singular point.

Let R be a commutative Noetherian local ring with residue field k. Let $\mathcal{X}$ be a resolving subcategory of finitely generated R-modules. This paper mainly studies when $\mathcal{X}$ contains k or consists of totally reflexive modules. It is proved that $\mathcal{X}$ does so if $\mathcal{X}$ is closed under cosyzygies. A conjecture of Dao and Takahashi is also shown to hold in several cases.

We prove the orbifold version of Zvonkine’s r-ELSV formula in two special cases: the case of $\mathit{r}=2$ (completed 3-cycles) for any genus $\mathit{g}\ge 0$ and the case of any $\mathit{r}\ge 1$ for genus $\mathit{g}=0$.

We prove that for any contact 3-manifold supported by a spinal open book decomposition with planar pages, there is a universal bound on the Euler characteristic and signature of its minimal symplectic fillings. The proof is an application of the spine removal surgery operation recently introduced in joint work of the authors with Van Horn-Morris [LVWa].

We study Deraux’s nonarithmetic orbifold ball quotient surfaces obtained as birational transformations of a quotient X of a particular Abelian surface A. Using the fact that A is the Jacobian of the Bolza genus 2 curve, we identify X as the weighted projective plane $\mathbb{P}(1,3,8)$. We compute the equation of the mirror M of the orbifold ball quotient $(\mathit{X},\mathit{M})$, and by taking the quotient by an involution we obtain an orbifold ball quotient surface with mirror birational to an interesting configuration of plane curves of degrees 1, 2, and 3. We also exhibit an arrangement of four conics in the plane that provides the above-mentioned ball quotient orbifold surfaces.