# Questions tagged [schemes]

The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1). It was finalized by Alexandre Grothendieck, during the 1950s and the 1960s.

468 questions
Filter by
Sorted by
Tagged with
320 views

### Mistake in Hartshorne's Exercise II.1.1?

This is really an elementary question, but let me state it. Exercise 1.1 of the second Chapter of Hartshorne's Algebraic Geometry ask to prove that the sheaf associated to the presheaf sending every ...
328 views

### Finite Picard group

Does there exist a connected scheme, smooth, proper, and positive-dimensional over $\mathbb{C}$ with finite Picard group? Note that Picard group has cardinality$>1$. Also note that this can not ...
272 views

### Affine open with irreducible complement

Let $X$ be an integral Noetherian separated scheme. Under what conditions can we find a non-empty affine open in $X$ whose complement is irreducible?
216 views

### Map to a given vector bundle from a split vector bundle

Let $X$ be a connected scheme, smooth and proper over $\mathbb{C}$. Let $F$ be a locally free $\mathcal{O}_X$-module of finite rank $r>1$. Suppose on a non-empty affine open $U\subset X$ whose ...
122 views

### Size of the ring of functions on open subschemes

This question consists of two related sub-questions. Let $X$ be a Noetherian integral affine scheme. Under what assumptions on $X$ does every open subscheme of $X$ have a Noetherian ring of global ...
243 views

### Do codimension 1 subsets of a scheme cover it?

Let $X$ be an irreducible scheme. A point $p\in X$ is said to have codimension $n\in\mathbb{Z}_{\geq 0}\cup \{\infty\}$ if $\overline{\{p\}}$ has codimension $n$. Is it true that any point of positive ...
193 views

### Schemes with no finite morphisms onto themselves

Let $X$ be an integral Noetherian scheme that has no automorphisms other than the identity. Is it possible to characterize (in a not completely tautological way) $X$ such that there is no finite ...
409 views

### Map of coherent sheaves inducing isomorphism on the stalks at the generic point

Let $f:X\rightarrow Y$ be a finite morphism between Noetherian integral schemes that is surjective on the underlying topological spaces. Does there exist an integer $n>0$ and a coherent $O_X$-...
78 views

### depth and extension of sections

Let $S$ be an affine scheme, $X$ smooth affine over $S$ and $U$ an open subset of $X$, fiberwise of codimension at least two. Suppose that we have a function on $U$, can we extend it to $X$?
46 views

### Homeomorphisms coinciding on closed irreducible subsets

Let $f_1$, $f_2:X\rightarrow Y$ be two homeomorphisms of sober spaces. Assume that for any closed irreducible subset $Z\subset X$, we have $f_1(Z)=f_2(Z)$. In particular, $f_1$ and $f_2$ coincide on ...
85 views

230 views

### Schemes monomorphing into affine scheme of dimension 1

Let $Y$ be an affine scheme of Krull dimension 1. Let $X\rightarrow Y$ be a monomorphism in the category of schemes. If $X$ is connected, is $X$ necessarily affine? What if we assume that $Y$ is a ...
431 views

### Voisin examples in $p$-adic geometry

Let $K$ be an algebraic closure of p-adic rationals. Does there exist a proper smooth rigid-analytic variety over $K$ whose etale homotopy type is not isomorphic to etale homotopy type of a proper ...
126 views

### Embedding smooth proper schemes into smooth proper schemes

Do there exist connected proper smooth $\mathbb{C}$-schemes $X_i$ ($\forall i\in \mathbb{Z}_{>0}$) with $\mathrm{dim}_{\mathbb{C}}X_i=i$ such that $X_i$ admits an immersion into $X_{i+1}$ and any ...
217 views

### Weil homotopy theory

In algebraic geometry, we have something called Weil cohomology theories, which formalize the notion of a "good" cohomology theory of smooth projective varieties. I believe that for $l$-adic ...
408 views

### Is there an analogue of projective spaces for proper schemes?

Does there exist a countable set of connected proper smooth $\mathbb{C}$-schemes such that any connected proper smooth $\mathbb{C}$-scheme admits a $\mathbb{C}$-immersion into one of them?
209 views

### Sufficient condition for quasi-compactness of scheme morphism

I wonder what kind of conditions on a morphism of schemes imply, in a non-trivial fashion, quasi-compactness of the morphism. Some examples Finiteness of surjective etale morphisms Is a universally ...
174 views

57 views

### Morphisms such that the inverse image of every affine open is contained in an affine open

Is there a name/description in standard terms of the class of morphisms of schemes defined by the following property: the inverse image of any affine open is contained in an affine open? It should ...
180 views

### Given a semi-abelian scheme, is the set of points such that the fibres are abelian varities open?

Let $\pi:\mathcal{A}\rightarrow C$ be a semi-abelian scheme, i.e. $\mathcal{A}$ is a smooth separated commutative group scheme over $C$ via $\pi$ with geometrically connected fibres, such that each ...
90 views

### Global functions on a flat proper family

Let $R$ be an integral domain. Let $f:X\rightarrow \mathrm{Spec}\,R$ be a flat proper morphism of schemes. Is it possible that $O_X(X)$ is not a flat $R$-module?
211 views

### Irreducible Smooth Proper one-dimensional Schemes isomorphic to $\mathbb{P}^1$

I have a curious question about an argument/hint given in following thread: https://math.stackexchange.com/questions/3062986/irreducible-smooth-proper-one-dimensional-mathbbz-schemes The OP asked if ...
116 views

### Projective subvarieties of blow-ups of affine varieties

Let $X$ be an integral affine scheme of finite type over a field $k$. Let $Y\subset X$ be an integral closed subscheme of codimension $n>0$. We blow up $X$ along $Y$ and get a $k$-scheme $X'$. Is ...
397 views

### Why is, for a group scheme of finite type, “smooth” (resp. irreducible) equivalent to “geometrically reduced” (resp. geometrically irreducible)?

I have some questions about two statements from Bosch's "Algebraic Geometry and Commutative Algebra" about algebraic varieties (page 479). Since I still don't have the permission to add images I quote ...