# Tagged Questions

**12**

votes

**2**answers

304 views

### Existence of local sections

I would like to know when the property of "having a section" for a morphism of varieties in characteristic $0$ can be detected by spreading out to characteristic $p$.
Take a number field $K$, and let ...

**1**

vote

**0**answers

86 views

### Benchmark problems for computing rational points on varieties

Are there standard benchmark problem sets used for empirically evaluating algorithms designed for computing rational points on (various classes of) algebraic varieties?
If so, could you please point ...

**3**

votes

**2**answers

118 views

### Lattice-point-free buffers around circles

Let $C(r)$ be the origin-centered circle of radius $r$,
and let $\beta(r)$ be the exterior buffer around $C(r)$:
the distance from $C(r)$ to the closest lattice point exterior to $C(r)$:
...

**7**

votes

**0**answers

141 views

### Lattice radial-step (ratchet) spirals

(30Oct13: Now solved; see Addendum.)
Define a curve, a ratchet spiral, $S(r_0,\epsilon)$ as follows, where $r_0 > 0$ and $\epsilon < 1$.
$S(r_0,\epsilon)$ begins with the arc ...

**0**

votes

**0**answers

96 views

### Existence of a curve with no points over finite separable field extensions

Does there exist a field $K$, and a smooth projective geometrically connected curve $C$ over $K$ such that, for all finite separable field extensions $L/K$ the curve $C$ has no $L$-rational points?
I ...

**9**

votes

**2**answers

299 views

### Rational points on circular spirals

Is it the case that every unit-radius circular spiral,
$$x = \cos(t)$$
$$y = \sin(t)$$
$$z = c \cdot t$$
for $c \in \mathbb{R}^+$
is dense in rational-coordinate points
(i.e., all three coordinates ...

**1**

vote

**1**answer

204 views

### Tricks to produce examples of hypersurfaces with index greater than $1$

Recall, index of an algebraic scheme $X$ is defined to be the greatest common divisor of the degrees of the space of zero cycles on $X$. I am interested in examples of hypersurfaces in ...

**6**

votes

**1**answer

334 views

### Simple field extension and rational points

Let $F$ be an infinite field and $f$ a homogeneous form on $F$ such that $f$ has no non-trivial zero in $F$. Let $F'$ be a finite extension of $F$ such that $f$ has a non-trivial zero in $F'$. Is it ...

**7**

votes

**5**answers

1k views

### Rational points on a sphere in $\mathbb{R}^d$

Call a point of $\mathbb{R}^d$ rational if all its
$d$ coordinates are rational numbers.
Q1.
Is the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$
dense in rational points, i.e., does $S$ ...

**7**

votes

**1**answer

266 views

### Varieties with infinitely many etale covers and rational points

Let $X$ be a (smooth projective geometrically connected) variety over a field $k$.
Consider the set Et$(X,k)$ of finite etale covers $Y\to X$ over $k$, with $Y$ geometrically connected over $k$.
...

**13**

votes

**2**answers

471 views

### Existence of points on varieties which avoid a given number field.

Let C be a geometrically integral curve over a number field K and let K' be a number field containing K. Does there exist a number field L containing K such that
$L \cap K' = K$, and
$C(L) \neq ...

**6**

votes

**1**answer

263 views

### Rational points on smooth compactifications

Let $X$ be as smooth variety over a field $k$ of characteristic $0$.
Consider the following statements:
The variety $X$ has no $k((t))$-rational points.
No smooth compactification of $X$ has a ...

**3**

votes

**1**answer

331 views

### Brauer-Manin obstruction and Hasse principle

I am looking for
varieties without $\mathbf{Q}$-rational points where the absence can be explained by the Brauer-Manin obstruction, but not by the absence of adelic points
varieties without ...

**8**

votes

**1**answer

627 views

### Rational points à la Chabauty-Coleman

I have been trying to learn the method of Chabauty and Coleman to find rational points on curves; I have been reading an exposition by McCallum and Poonen which was pointed out to me by Emerton in ...