We enumerate the surfaces of degree d in ${\mathbb{P}}^3$ having a singular line of order k and passing through δ generic points (where δ is the dimension of moduli space of such surfaces).

In this paper we revisit Ribenboim’s notion of higher derivations of modules and relate it to the recent work of de Fernex and Docampo on the sheaf of differentials of the arc space. In particular, we derive their formula for the Kähler differentials of the Hasse–Schmidt algebra as a consequence of the fact that the Hasse–Schmidt algebra functors commute.

We prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen ${\mathit{d}}_{\mathit{t}}$ invariant of braid closures. Focusing on the case of 3-braids, we compute the Rasmussen s-invariant and the annular Rasmussen ${\mathit{d}}_{\mathit{t}}$ invariant of all 3-braid closures. As a corollary, we show that the vanishing/non-vanishing of the ψ invariant is entirely determined by the s-invariant and the self-linking number for 3-braid closures.

Regular semisimple Hessenberg varieties are smooth subvarieties of the flag variety, and their examples contain the flag variety itself and the permutohedral variety, which is a toric variety. We give a complete classification of Fano and weak Fano regular semisimple Hessenberg varieties in type A in terms of combinatorics of Hessenberg functions. In particular, we show that if the anticanonical bundle of a regular semisimple Hessenberg variety is nef, then it is in fact nef and big.

In this paper, we are concerned with the well-posedness of Vlasov–Poisson equation near vacuum in weighted fractional Sobolev space ${\mathit{H}}^{\mathit{s},\mathit{p}}(\mathit{w})$, that is, Bessel potential space. The difficulty lies in the estimates of the electronic term ${\nabla }_{\mathit{x}}\mathit{\varphi }$. To overcome it, we establish the ${\mathit{L}}^{\mathit{p}}$-${\mathit{L}}^{\mathit{q}}$ estimate and take advantage of commutator estimates. The ${\mathit{L}}^{\mathit{p}}$-version energy method and pseudo-differential operator techniques are also introduced.

Rees-like algebras have played a major role in settling the Eisenbud–Goto conjecture. This paper concerns the structure of the canonical module of the Rees-like algebra and its class groups. Via an explicit computation based on linkage, we provide an explicit and surprisingly well-structured resolution of the canonical module in terms of a type of double-Koszul complex. Additionally, we give descriptions of both the divisor class group and the Picard group of a Rees-like algebra.

We prove that for each positive integer n, there exists a positive number ${\mathit{ϵ}}_{\mathit{n}}$ such that n-dimensional toric quotient singularities satisfy the ACC for MLDs on the interval $(0,{\mathit{ϵ}}_{\mathit{n}})$. In the course of the proof, we show a geometric Jordan property for finite automorphism groups of affine toric varieties.

We show the existence of several new infinite families of polynomially-growing automorphisms of free groups whose mapping tori are CAT(0) free-by-cyclic groups. Such mapping tori are thick, and thus not relatively hyperbolic. These are the first families comprising infinitely many examples for each rank of the nonabelian free group; they contrast strongly with Gersten’s example of a thick free-by-cyclic group which cannot be a subgroup of a CAT(0) group.

We study the monodromies and the limit mixed Hodge structures of families of complete intersection varieties over a punctured disk in the complex plane. For this purpose, we express their motivic nearby fibers in terms of the geometric data of some Newton polyhedra. In particular, the limit mixed Hodge numbers and some part of the Jordan normal forms of the monodromies of such a family will be described very explicitly.

It is well known that adding a general boundary creates K-stability. As an application, we reprove the product theorem for delta invariants of Fano varieties.