Related papers: Which quartic double solids are rational?
We give uniform upper bounds for the number of rational points of height at most $B$ on non-singular complete intersections of two quadrics in $\mathbb{P}^3$ defined over $\mathbb{Q}$. To do this, we combine determinant methods with descent…
We propose and compare various techiques available to produce smooth cubic hypersurfaces over a non-algebraically-closed field which have rational points but which are not stably rational over their ground field.
By leveraging the physics of the Higgs branch, we argue that the conformal central charges $a$ and $c$ of an arbitrary 4d $N=2$ superconformal field theory (SCFT) are rational numbers. Our proof of the rationality of $c$ is conditioned on a…
We show that the maximal number of (real) lines in a (real) nonsingular spatial quartic surface is 64 (respectively, 56). We also give a complete projective classification of all quartics containing more than 52 lines: all such quartics are…
One of the greatest achievements of Greek mathematics is the proof that the square root of 2 is irrational. It has not been thought that the Babylonians appreciated the concept of irrationality and certainly that they did not prove that the…
We prove the existence of a family $\mathcal{X}\rightarrow B$ of smooth projective fourfolds, such that the very general fiber $\mathcal{X}_t$ is not stably rational (a fortiori not rational), but some special fibers $\mathcal{X}_t$ are…
We give some bounds on the numbers of rational points on abelian varieties and jacobians varieties over finite fields. The main result is that we determine the maximum and minimum number of rational points on jacobians varieties of…
Among geometrically rational surfaces, del Pezzo surfaces of degree two over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for k-rational curves on these surfaces, we…
There exists "a square problem": in a unit square is there a point with four rational distances to the vertices? This problem is still regarded as unproved. Yang showed proofs for several special cases of the square problem. By the…
We show that even dimensional Fermat cubic hypersurfaces are rational over any field of characteristic different from three by producing explicit rational parametrizations given by polynomials of low degree. As a byproduct of our…
This paper suggests a new approach to questions of rationality of threefolds based on category theory. Following M. Ballard, D. Favero, L. Katzarkov (ArXiv:1012.0864) and D. Favero, L. Katzarkov (Noether--Lefschetz Spectra and Algebraic…
Let $\mathcal{Q}$ be an irreducible quartic with two nodes and one cusp as its singularities and let $\mathcal{C}$ be a conic such that the intersection multiplicity at each point of $\mathcal{C} \cap \mathcal{Q}$ is even and $\mathcal{C}…
We give a conjectural formula for the characteristic number of rational cuspidal curves in the projective plane by extending the idea of Kontsevich's recursion formula (namely, pulling back the equality of two divisors in the four pointed…
Rational pairs generalize the notion of rational singularities to reduced pairs $(X,D)$. In this paper we deal with the problem of determining whether a normal variety $X$ has a rationalizing divisor, i.e. a reduced divisor $D$ such that…
If the cosine of a rational multiple of $\pi$ is a rational number then it is an integral multiple of $\frac12$. For this fact, we give a proof accessible to an interested school student. We then discuss which quadratic and cubic…
The moduli spaces of trigonal curves are proven to be rational when the genus is divisible by 4.
Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over $\mathbb{Q}$ that contains a conic defined over $\mathbb{Q}$.
A set of $m$ distinct nonzero rationals $\{a_1,a_2,\ldots,a_m\}$ such that $a_ia_j+1$ is a perfect square for all $1\leq i<j\leq m$, is called a rational Diophantine $m$-tuple. It is proved recently that there are infinitely many rational…
In this note, we use integral binary cubic forms to study the rational cube sum problem. We prove (unconditionally) that for any positive integer $d$, infinitely many primes in each of the residue classes $ 1 \pmod {9d}$ as well as $ -1…
A famous problem posed by Diophantus was to find sets of distinct positive rational numbers such that the product of any two is one less than a rational square. Some sets of six such numbers are presented and the computational algorithm…