Search results for "Projective space"
showing 2 items of 52 documents
On the arithmetic of a family of degree-two K3 surfaces
2018
Let $\mathbb{P}$ denote the weighted projective space with weights $(1,1,1,3)$ over the rationals, with coordinates $x,y,z,$ and $w$; let $\mathcal{X}$ be the generic element of the family of surfaces in $\mathbb{P}$ given by \begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*} The surface $\mathcal{X}$ is a K3 surface over the function field $\mathbb{Q}(t)$. In this paper, we explicitly compute the geometric Picard lattice of $\mathcal{X}$, together with its Galois module structure, as well as derive more results on the arithmetic of $\mathcal{X}$ and other elements of the family $X$.
The ACM property for unions of lines in P1×P2
2021
This paper examines the Arithmetically Cohen-Macaulay (ACM) property for certain codimension 2 varieties in P1×P2 called sets of lines in P1×P2 (not necessarily reduced). We discuss some obstacles to finding a general characterization. We then consider certain classes of such curves, and we address two questions. First, when are they themselves ACM? Second, in a non-ACM reduced configuration, is it possible to replace one component of a primary (prime) decomposition by a suitable power (i.e. to “fatten” one line) to make the resulting scheme ACM? Finally, for our classes of such curves, we characterize the locally Cohen-Macaulay property in combinatorial terms by introducing the definition …