Inhomogeneous linear equation/Geometric situation/description

We consider a linear homogeneous equation

$f_{1}t_{1}+\cdots +f_{n}t_{n}=0$

and also a linear inhomogeneous equation

$f_{1}t_{1}+\cdots +f_{n}t_{n}=f_{0},$

where ${}f_{1},\ldots ,f_{n},f_{0}$ are elements in a field ${}K$. The solution set to the homogeneous equation is a vector space ${}V$ over ${}K$ of dimension ${}n-1$ or ${}n$ (if all ${}f_{i}$ are ${}0$). For the solution set ${}T$ of the inhomogeneous equation there exists an action

$V\times T\longrightarrow T,\,(v,t)\longmapsto v+t,$

and if we fix one solution ${}t_{0}\in T$

(supposing that one solution exists), then there exists a bijection

$V\longrightarrow T,\,v\longmapsto v+t_{0}.$

Suppose now that ${}X$ is a geometric object (a topological space, a manifold, a variety, the spectrum of a ring) and that

$f_{1},\ldots ,f_{n},f_{0}\colon X\longrightarrow K$

are functions on ${}X$. Then we get the space

$T={\left\{(P,t_{1},\ldots ,t_{n})\mid f_{1}(P)t_{1}+\cdots +f_{n}(P)t_{n}=f_{0}(P)\right\}}\subseteq X\times K^{n}$

together with the projection to ${}X$. For a fixed point ${}P\in X$, the fiber of ${}T$ over ${}P$ is the solution set to the corresponding inhomogeneous equation. For ${}f_{0}=0$, we get a solution space

$V\longrightarrow X,$

where all fibers are vector spaces

(maybe of non-constant dimension) and where again ${}V$ acts on ${}T$. Locally, there are bijections ${}V\cong T$. Let

$U={\left\{Q\in X\mid f_{i}(Q)\neq 0{\text{ for at least one }}i\right\}}.$

Then ${}V{|}_{U}$ is a vector bundle and ${}T{|}_{U}$ is a ${}V{|}_{U}$-principal fiber bundle.

${}T$ is fiberwise an affine space over the base and locally an affine space over ${}U$, so locally it is an easy object. We are interested in global properties of ${}T$ and of ${}T{|}_{U}$.