For families of $\mathit{K}3$ surfaces, we establish a sufficient criterion for real or complex multiplication. Our criterion is arithmetic in nature. It may show, at first, that the generic fibre of the family has a nontrivial endomorphism field. Moreover, the endomorphism field does not shrink under specialization. As an application, we present two explicit families of $\mathit{K}3$ surfaces having real multiplication by $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{5})$, respectively.

As a generalization of the ideals of star configurations of hypersurfaces, we consider the a-fold product ideal ${\mathit{I}}_{\mathit{a}}({\mathit{f}}_1^{{\mathit{m}}_1}\cdots {\mathit{f}}_{\mathit{s}}^{{\mathit{m}}_{\mathit{s}}})$ when ${\mathit{f}}_1,\dots ,{\mathit{f}}_{\mathit{s}}$ is a sequence of n-generic forms and $1\le \mathit{a}\le {\mathit{m}}_1+\cdots +{\mathit{m}}_{\mathit{s}}$. Firstly, we show that this ideal has complete intersection quotients when these forms are of the same degree and essentially linear. Then, we study its symbolic powers while focusing on the uniform case with ${\mathit{m}}_1=\cdots ={\mathit{m}}_{\mathit{s}}$. For large a, we describe its resurgence and symbolic defect. And for general a, we also investigate the corresponding invariants for meeting-at-the-minimal-components version of symbolic powers.

We generalize the classical semicontinuity theorem for GIT (semi)stable loci under variations of linearizations to a relative situation of an equivariant projective morphism $\mathit{X}\to \mathit{S}$ over an affine base S. As an application to moduli problems, we consider degenerations of Hilbert schemes and give a conceptual interpretation of the (semi)stable loci of the degeneration families constructed in [GHH19].

Let L be a nonnegative self-adjoint operator acting on ${\mathit{L}}^2(\mathit{X})$, where X is a space of homogeneous type of dimension n. Suppose that the heat kernel of L satisfies a Gaussian upper bound. It is known that the operator ${(\mathit{I}+\mathit{L})}^{-\mathit{s}}{\mathit{e}}^{\mathit{i}\mathit{t}\mathit{L}}$ is bounded on ${\mathit{L}}^{\mathit{p}}(\mathit{X})$ for $\mathit{s}>\mathit{n}|1/2-1/\mathit{p}|$ and $\mathit{p}\in (1,\infty )$ (see, e.g., [7, 22, 33]). The index $\mathit{s}=\mathit{n}|1/2-1/\mathit{p}|$ was only obtained recently in [9, 10], and this range of s is sharp since it is precisely the range known in the case where L is the Laplace operator Δ on $\mathit{X}={\mathbb{R}}^{\mathit{n}}$ [30]. In this paper, we establish that for $\mathit{p}=1$, the operator ${(1+\mathit{L})}^{-\mathit{n}/2}{\mathit{e}}^{\mathit{i}\mathit{t}\mathit{L}}$ is of weak type $(1,1)$, that is, there is a constant C, independent of t and f, such that

(for $\mathit{\lambda }>0$ when $\mathit{\mu }(\mathit{X})=\infty$ and $\mathit{\lambda }>\mathit{\mu }{(\mathit{X})}^{-1}\Vert \mathit{f}{\Vert }_{{\mathit{L}}^1(\mathit{X})}$ when $\mathit{\mu }(\mathit{X})<\infty$). Moreover, we also show that the index $\mathit{n}/2$ is sharp when L is the Laplacian on ${\mathbb{R}}^{\mathit{n}}$ by providing an example.

Our results are applicable to Schrödinger groups for large classes of operators including elliptic operators on compact manifolds, Schrödinger operators with nonnegative potentials and Laplace operators acting on Lie groups of polynomial growth or irregular nondoubling domains of Euclidean spaces.

We obtain a new bound on Weyl sums with degree $\mathit{k}\ge 2$ polynomials of the form $(\mathit{\tau }\mathit{x}+\mathit{c})\mathit{\omega }(\mathit{n})+\mathit{x}\mathit{n}$, $\mathit{n}=1,2,\dots$, with fixed $\mathit{\omega }(\mathit{T})\in \mathbb{Z}[\mathit{T}]$ and $\mathit{\tau }\in \mathbb{R}$, which holds for almost all $\mathit{c}\in [0,1)$ and all $\mathit{x}\in [0,1)$. We improve and generalize some recent results of Erdoǧan and Shakan (2019), whose work also shows links between this question and some classical partial differential equations. We extend this to more general settings of families of polynomials $\mathit{x}\mathit{n}+\mathit{y}\mathit{\omega }(\mathit{n})$ for all ${(\mathit{x},\mathit{y})\in [0,1)}^2$ with $\mathit{f}(\mathit{x},\mathit{y})=\mathit{z}$ for a set of $\mathit{z}\in [0,1)$ of full Lebesgue measure, provided that f is a Hölder function.

For a classical link, Milnor defined a family of isotopy invariants, called Milnor $\bar{\mathit{\mu }}$-invariants. Recently, Chrisman extended Milnor $\bar{\mathit{\mu }}$-invariants to welded links by a topological approach. The aim of this paper is showing that Milnor $\bar{\mathit{\mu }}$-invariants can be extended to welded links by a combinatorial approach. The proof contains an alternative proof of the invariance of the original $\bar{\mathit{\mu }}$-invariants for classical links.

We prove that integral projective curves with Cohen–Macaulay singularities over an algebraically closed field of characteristic zero are determined by their triangulated category of perfect complexes. This partially extends a theorem of Bondal and Orlov to the case of singular projective varieties and in particular shows that all integral complex projective curves are determined by their category of perfect complexes.

We study truncated point schemes of connected graded algebras as families over the parameter space of varying relations for the algebras, proving that the families are flat over the open dense locus where the point schemes achieve the expected (i.e., minimal) dimension.

When the truncated point scheme is zero-dimensional, we obtain its number of points counted with multiplicity via a Chow ring computation. This latter application in particular confirms a conjecture of Brazfield that a generic two-generator two-relation algebra has seventeen truncated point modules of length six.

Every L-space knot is fibered and strongly quasi-positive, but this does not hold for L-space links. In this paper, we use the so-called H-function, which is a concordance link invariant, to introduce a subfamily of fibered strongly quasi-positive L-space links. Furthermore, we present an infinite family of L-space links that are not quasi-positive.