Let $j:U\rightarrow X$ an open immersion between k-schemes of finite type and $f:X\rightarrow S$ a surjective k-morphism of finite type. We suppose that $f\circ j:U\rightarrow S$ …

Let $f:X\to S$ be a "nice" morphism of "nice" schemes. Let $L$ be an ample line bundle on $X$. When is $\det f_\ast L$ also ample? A "nice" morphism could be anything from "finit …

Let a cartesian diagram Let $X'\rightarrow X$ be a rational resolution of singularities of $k$-schemes of finite type and $Y$ a closed subscheme. Let $Y'\rightarrow Y$ be the bas …

I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of e …

Toen-Vaquié construct a category of schemes relative to some complete cocomplete closed symmetric monoidal category $C$. Affine schemes correspond by definition 1:1 to commutative …

Let $A$ be a (commutative) ring. A family $(B_i)_{i\in I}$ of $A$-algebras is said to be an fppf cover if it satisfies three properties: (1) each $B_i$ is flat as an $A$-module, (2 …

I can prove a certain statement for any scheme that can be presented as the limit of an essentially affine (filtering) projective system of smooth varieties over a perfect field su …

Let $S\cong \varprojlim S_i$, where $S$ and all $S_i$ are separated regular excellent of finite Krull dimension. Let $Z$ be a closed regular subscheme of $S$. As Theorem 8.8.2 of E …

I've been looking many places, but everything I find seems to either talk about (a) varieties or (b) extremely general situations with dualizing complexes. As I am not in the situa …

In my preprint I propose to call $X/S$ quasi-smooth if $X$ can be embedded into a smooth $X'/S$. Does this sound fine? Upd. So, smoothly embeddable is better? Is it ok to call a m …

It seems to be well-known (see http://mathoverflow.net/questions/201/is-there-an-example-of-a-variety-over-the-complex-numbers-with-no-embedding-into) that a general finite type $S …

(This was originally asked on math.stackexchange, but didn't get any responses. I figured it might be worthwhile to move it here and try again.) This paper gives a proof that the …

Let $S$ be a scheme. A smooth curve over $S$ is a smooth projective $S$-scheme of relative dimension $1$ with geometrically connected fibres. A semi-stable curve over $S$ is a f …

Let $S$ be an integral 1-dimensional scheme with function field $K$. Let $E$ be an elliptic curve over $K$. The torsion of $E$ over $K$ is not necessarily finite. As an example co …

Let $A$ be a commutative ring and $n \in \mathbb{N}$. What is the minimal number $e_A(n)$ of generators of the $A$-algebra $A^n$? Here is what I already know (I can add proofs if n …