We generalize Lempert’s and Poletsky’s works [11, 14] on the description of extremal discs for the Kobayashi metric to a higher order setting with $k$-stationarity condition introduced in [1].

When studying subgroups of $\mathsf{Out}\left(F_n\right)$, one often replaces a given subgroup $\mathcal{H}$ with one of its finite index subgroups ${\mathcal{H}}_0$ so that virtual properties of $\mathcal{H}$ become actual properties of ${\mathcal{H}}_0$. In many cases, the finite index subgroup is ${\mathcal{H}}_0=\mathcal{H}\cap {IA}_n(\mathbb{Z}/3)$. For which properties is this a good choice? Our main theorem states that being abelian is such a property. Namely, every virtually abelian subgroup of ${IA}_n(\mathbb{Z}/3)$ is abelian.

We perform a variation of geometric invariant theory stability analysis for 2nd Hilbert points of bi-log-canonically embedded pointed curves of genus $2$. As a result, we give a GIT construction of the log canonical models ${\overline{M}}_{2,1}\left(\alpha \right)$ for $\alpha =2/3\pm ϵ$ and obtain a VGIT presentation of the second flip in the Hassett–Keel program for the moduli space of pointed genus $2$ curves.

For blow-ups of threefolds along $(-1,-1)$-curves, we use the degeneration formula and the absolute/relative correspondence to obtain some closed blow-up formulae for Gromov–Witten invariants and generalized BPS numbers.

For any positive integer $k$, we define ${\Gamma }^{\ast }\left(k\right)$ to be the smallest number $s$ such that every diagonal form $a_1{x}_1^k+a_2{x}_2^k+\cdots +a_s{x}_s^k$ in $s$ variables with integer coefficients must have a nontrivial zero in every $p$-adic field ${\mathbb{Q}}_p$. An old conjecture of Norton is that we should have ${\Gamma }^{\ast }\left(k\right)\equiv 1\left(\text{mod}k\right)$ for all $k$. For many years, ${\Gamma }^{\ast }\left(8\right)=39$ was the only known counterexample to this conjecture, and in recent years two more counterexamples have been found. In this article, we produce infinitely many counterexamples to Norton’s conjecture.

Let $L$ be a line bundle on a scheme $X$, proper over a field. The property of $L$ being nef can sometimes be “thickened”, allowing reductions to positive characteristic. We call such line bundles arithmetically nef. It is known that a line bundle $L$ may be nef, but not arithmetically nef. We show that $L$ is arithmetically nef if and only if its restriction to its stable base locus is arithmetically nef. Consequently, if $L$ is nef and its stable base locus has dimension $1$ or less, then $L$ is arithmetically nef.

It is a well-known fact that endomorphisms of $B\left(H\right)$ are intimately connected with families of mutually orthogonal isometries, that is, with representations of the so-called Toeplitz $C^{\ast }$-algebras. In this paper we consider a natural generalization of this connection between the representation theory of certain $C^{\ast }$-algebras associated with graphs and endomorphisms of certain von Neumann subalgebras of $B\left(H\right)$. Our primary results give criteria by which it may be determined if two representations give rise to equal or conjugate endomorphisms.

We study some particular loci inside the moduli space ${\mathcal{M}}_g$, namely the bielliptic locus (i.e. the locus of curves admitting a $2:1$ cover over an elliptic curve $E$) and the bihyperelliptic locus (i.e. the locus of curves admitting a $2:1$ cover over a hyperelliptic curve $C\text{'}$, $g\left(C\text{'}\right)\ge 2$). We show that the bielliptic locus is not a totally geodesic subvariety of ${\mathcal{A}}_g$ if $g\ge 4$ (whereas it is for $g=3$, see [18]) and that the bihyperelliptic locus is not totally geodesic in ${\mathcal{A}}_g$ if $g\ge 3g\text{'}$. We also give a lower bound for the rank of the second Gaussian map at the generic point of the bielliptic locus and an upper bound for this rank for every bielliptic curve.

We study the bi-graded Hilbert function of ideals of general fat points with same multiplicity in ${\mathbb{P}}^1\times {\mathbb{P}}^1$. Our first tool is the multiprojective-affine-projective method introduced by the second author in previous works with A. V. Geramita and A. Gimigliano where they solved the case of double points. In this way, we compute the Hilbert function when the smallest entry of the bi-degree is at most the multiplicity of the points. Our second tool is the differential Horace method introduced by J. Alexander and A. Hirschowitz to study the Hilbert function of sets of fat points in standard projective spaces. In this way, we compute the entire bi-graded Hilbert function in the case of triple points.

We give a group cohomological description of the Čech cohomology of the Bowditch boundary of a relatively hyperbolic group pair, generalizing a result of Bestvina–Mess about hyperbolic groups. In the case of a relatively hyperbolic Poincaré duality group pair, we show that the Bowditch boundary is a homology manifold. For a three-dimensional Poincaré duality pair, we recover the theorem of Tshishiku–Walsh stating that the boundary is homeomorphic to a two-sphere.