---
title: "Public Solutions and Coverage of Worksheet 22"
stable_id: br-bgk-2019-w22-solutions
language: en
source_course: "Kurs:Bündel, Garben und Kohomologie (Osnabrück 2019-2020)"
source_author: "Holger Brenner"
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solution_ex19_upstream_title: "Fermat-Kubik/Projektiv/Disjunkte Geraden/Aufgabe/Lösung"
solution_ex19_upstream_pageid: 115918
solution_ex19_upstream_revid: 1089477
solution_ex19_upstream_timestamp: "2026-05-31T10:35:22Z"
solution_ex19_mediawiki_sha1: 4394016a5f0bad50ae7047a33f765ed1650e2e54
solution_ex19_frozen_revision_contributor: "Arbota"
solution_ex19_source_url: "https://de.wikiversity.org/w/index.php?oldid=1089477"
exercise_count: 20
public_solution_count: 1
public_solution_numbers: "19"
negative_public_solution_count: 19
negative_solution_numbers: "1-18, 20"
license: "Frozen semantic course text and this translation: CC BY-SA 4.0."
non_endorsement: "This independent English edition does not imply endorsement by the author, Wikiversity, the Wikimedia Foundation, or any source institution."
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---

```{=latex}
\clearpage
```

# Public Solutions and Coverage of Worksheet 22 {#br-bgk-2019-w22-solutions}

The candidate-title checks recorded by QA, based on the sequence of
20 exercises in the frozen worksheet revision, found exactly one public
solution page: the solution to Exercise 22.19. The results were negative
for Exercises 22.1--22.18 and 22.20. The absence of a public solution
page is not replaced by an invented solution.

## Solution to Exercise 22.19 {#br-bgk-2019-w22-sol-ex19}

We write the defining equation as

$$
\begin{aligned}
F
&=X^3-Z^3+Y^3-W^3\\
&=(X-Z)(X-\zeta Z)(X-\zeta^2Z)
 +(Y-W)(Y-\zeta W)(Y-\zeta^2W),
\end{aligned}
$$

where $\zeta$ is a third root of unity. Thus, for example,

$$
F\in(X-Z,Y-W)
$$

and

$$
F\in(X-\zeta Z,Y-\zeta W).
$$

Hence

$$
L_1=V_+(X-Z,Y-W),
\qquad
L_2=V_+(X-\zeta Z,Y-\zeta W)
\subseteq V_+(F).
$$

As intersections of two distinct projective planes, $L_1$ and $L_2$
are lines. Their intersection is

$$
\begin{aligned}
L_1\cap L_2
&=V_+(X-Z,Y-W),\ L_2\cap V_+(X-\zeta Z,Y-\zeta W)\\
&=V_+(X-Z,Y-W,X-\zeta Z,Y-\zeta W)\\
&=V_+(X-Z,X-\zeta Z,Y-W,Y-\zeta W)\\
&=V_+(X,Z,Y,W)\\
&=\varnothing.
\end{aligned}
$$

> **Editorial note - first line of the intersection chain.** The first line
> of the source display prints a comma, repeats $L_2$, and then writes
> a second intersection. This edition preserves that line as in the source;
> the subsequent lines give the combined ideal yielding the empty intersection.
> The root $\zeta$ must be primitive, so $\zeta\ne1$; otherwise the two
> lines coincide and the factorisation is invalid in characteristic
> different from $3$. Such a root exists under the exercise's hypotheses.

## Negative candidate-check results {#br-bgk-2019-w22-solutions-negative}

Those candidate checks found no public solution page for Exercises 22.1,
22.2, 22.3, 22.4, 22.5, 22.6, 22.7, 22.8, 22.9, 22.10, 22.11,
22.12, 22.13, 22.14, 22.15, 22.16, 22.17, 22.18, or 22.20.
This statement records the candidate checks; it does not assert that
mathematical solutions do not exist.