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David McKinnon

Publications and source records attributed to David McKinnon.

28 records · Page 2Linked to original sources
arXiv2015

Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties

In this paper, we associate an invariant $α_{x}(L)$ to an algebraic point $x$ on an algebraic variety $X$ with an ample line bundle $L$. The invariant $α$ measures how well $x$ can be approximated by rational points on $X$, with respect to the height function associated to $L$. We show that this invariant is closely related to the Seshadri constant $ε_{x}(L)$ measuring local positivity of $L$ at $x$, and in particular that Roth's theorem on $\mathbf{P}^1$ generalizes as an inequality between these two invariants valid for arbitrary projective varieties.

math.AG↗
arXiv2011

Ideals of degree one contribute most of the height

Let $k$ be a number field, $f(x)\in k[x]$ a polynomial over $k$ with $f(0)\neq 0$, and $Ø_{k,S}^*$ the group of $S$-units of $k$, where $S$ is an appropriate finite set of places of $k$. In this note, we prove that outside of some natural exceptional set $T\subset Ø_{k,S}^*$, the prime ideals of $Ø_k$ dividing $f(u)$, $u\in Ø_{k,S}^*\setminus T$, mostly have degree one over $\Q$; that is, the corresponding residue fields have degree one over the prime field. We also formulate a conjectural analogue of this result for rational points on an elliptic curve over a number field, and deduce our conjecture from Vojta's Conjecture. We prove this conjectural analogue in certain cases when the elliptic curve has complex multiplication.

math.NT↗
arXiv2010

Vojta's Conjecture implies the Batyrev-Manin Conjecture for $K3$ surfaces

Vojta's Conjectures are well known to imply a wide range of results, known and unknown, in arithmetic geometry. In this paper, we add to the list by proving that they imply that rational points tend to repel each other on algebraic varieties with nonnegative Kodaira dimension. We use this to deduce, from Vojta's Conjectures, conjectures of Batyrev-Manin and Manin on the distribution of rational points on algebraic varieties. In particular, we show that Vojta's Main Conjecture implies the Batyrev-Manin Conjecture for $K3$ surfaces.

math.NT↗
arXiv2010

K 1-6: an asymmetric planetary nebula with a binary central star

We present new imaging data and archival multiwavelength observations of the little studied emission nebula K 1-6 and its central star. Narrow-band images in H-alpha (+ [NII]) and [OIII] taken with the Faulkes Telescope North reveal a stratified, asymmetric, elliptical nebula surrounding a central star which has the colours of a late G- or early K-type subgiant or giant. GALEX ultraviolet images reveal a very hot subdwarf or white dwarf coincident in position with this star. The cooler, optically dominant star is strongly variable with a period of 21.312 +/- 0.008 days, and is possibly a high amplitude member of the RS CVn class, although an FK Com classification is also possible. Archival ROSAT data provide good evidence that the cool star has an active corona. We conclude that K 1-6 is most likely an old bona fide planetary nebula at a distance of ~1.0 kpc, interacting with the interstellar medium, and containing a binary or ternary central star. The observations and data analyses reported in this paper were conducted in conjunction with Year 11 high school students as part of an Australian Research Council Linkage Grant science education project, denoted Space To Grow, conducted jointly by professional astronomers, educational researchers, teachers, and high-school students.

astro-ph.SR↗
arXiv2009

Density of rational points on diagonal quartic surfaces

Let a,b,c,d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in PP^3 defined by ax^4+by^4+cz^4+dw^4=0. We prove that if V contains a rational point that does not lie on any of the 48 lines on V or on any of the coordinate planes, then the set of rational points on V is dense in both the Zariski topology and the real analytic topology.

math.AG↗
arXiv2008

K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces $X$ defined over a number field $k$, if there is an algebraic point on $X$ that lies on only finitely many rational curves, then there is an algebraic point on $X$ that lies on no rational curves. In particular, our theorem applies to a large class of elliptic $K3$ surfaces, which relates to a question posed by Bogomolov in 1981. We apply our results to construct an explicit algebraic point on a $K3$ surface that does not lie on any smooth rational curves.

math.AG↗
arXiv2006

A conjecture on rational approximations to rational points

In this paper, we examine how well a rational point P on an algebraic variety X can be approximated by other rational points. We conjecture that if P lies on a rational curve, then the best approximations to P on X can be chosen to lie along a rational curve. We prove this conjecture for a wide range of examples, and for a great many more examples we deduce our conjecture from Vojta's Main Conjecture.

math.NT↗
arXiv2002

Counting Rational Points on Ruled Varieties

In this paper, we prove a general result computing the number of rational points of bounded height on a projective variety $V$ which is covered by lines. The main technical result used to achieve this is an upper bound on the number of rational points of bounded height on a line. This upper bound is such that it can be easily controlled as the line varies, and hence is used to sum the counting functions of the lines which cover the original variety $V$.

math.AG↗
arXiv1999

Counting Rational Points on K3 Surfaces

For any algebraic variety $V$ defined over a number field $k$, and ample height function $H$ on $V$, one can define the counting function $N_V(B) = #{P\in V(k) \mid H(P)\leq B}$. In this paper, we calculate the counting function for Kummer surfaces $V$ whose associated abelian surface is the product of elliptic curves. In particular, we effectively construct a finite union $C = \cup C_i$ of curves $C_i$ on $V$ such that $N_{V-C}(B)\ll N_C(B)$; that is, $C$ is an accumulating subset of $V$. In the terminology of Batyrev and Manin, this amounts to proving that $C$ is the first layer of the arithmetic stratification of $V$.

math.AG↗