# 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.

653 questions
Filter by
Sorted by
Tagged with
130 views

### Is this blow-up a line bundle over the projective line

Let $R$ be the ring $\mathbb{C}[a,b,c,d]/(ac-b^2,bd-c^2,ad-bc)$. Let $I$ be the ideal of $R$ generated by $a,d$. Let $X=\operatorname{Spec}(R)$ and $\tilde{X}$ be the blow-up of the affine scheme $X$ ...
• 71
163 views

### Does a smooth relative curve $X/S$ embed into $\mathbb{P}^3_S$?

Suppose that $\pi:X\to S$ is a smooth projective morphism of relative dimension 1. If $S$ is the spectrum of an algebraically closed field, then it is known that $X$ embeds into $\mathbb{P}^3_S$. ...
• 1,033
1 vote
103 views

• 71
136 views

### Constructibility of the locus of points where the fiber is an isomorphism modulo nilpotents

Let $f: X \rightarrow S$ be a finitely presented morphism of schemes and let $$E = \{s \in S \mid \text{ X_s is a point with residue field \kappa(s) } \}$$ Is $E$ a constructible set? The basic ...
• 908
184 views

### Henselisation of normal rings (Milne's EC)

The usual way to define the Henselisation $A^h$ of a local ring $(A, \mathfrak{m})$ is by taking the direct limit $\varinjlim (B, q)$ over all etale neighborhoods of $A$ (i.e. pairs $(B,q)$ where $B$ ...
• 903
244 views

### Normal schemes and Serre's criterion

Serre's criterion says that for a scheme to be normal is equivalent to it being $R_1$ (i.e. regular in codimension $1$) and $S_2$ (i.e. regular functions on $X-Y$ extend to $Y$ if $Y$ has codimension ...
• 143
121 views

• 457
1 vote
86 views

### Intersection multiplicity in flat families of linear spaces

Let $X\subset\mathbb{P}^N$ be an irreducible projective variety and $\{H_t\}_{t\in \mathbb{C}^{*}}$ a family of $(k-2)$-dimensional linear subspaces of $\mathbb{P}^N$ intersecting $X$ in $k$ distinct ...
• 101
231 views

### Finite coverings by closed subschemes

Let $X$ be a scheme. Assume we have two closed subschemes $Y_1$, $Y_2$ that cover $X$ set-theoretically. Are there closed subschemes $Y'_1$, $Y'_2$ with the same underlying sets, such that the ...
• 18.7k
178 views

### Does a universal homeomorphism descend ampleness?

All schemes are quasicompact and separated. If $\pi:X\to Y$ is an affine morphism and $L$ is an ample invertible $\mathscr{O}_Y$-module, then $\pi^*L$ is ample on $X$; we say that $\pi$ descends ...
• 18.7k
88 views

### Definition of maximal order on an integral scheme

In the paper: The minimal model program for orders over surfaces by Daniel Chan and Colin Ingalls, they give the following definition. Let $Z$ be an integral normal scheme with quotient field $K$. An ...
• 123
187 views

### Is the "naive" version of Chevalley's theorem still true?

Reposting from math.se in case more people are interested here. Chevalley's theorem says that if $f \colon X \to Y$ is a morphism of finite presentation of schemes and $C \subset X$ is constructible, ...
1 vote
151 views

### Meaning of "cut out (scheme-theoretically)"

Let $V$ be a projectively normal closed subvariety of some projective space over an algebraically closed field $\mathbb{K}$. Let $R$ be the local ring at the vertex of the affine cone over $V$ ($R$ is ...
• 391
1 vote
99 views

### Arithmetic ampleness and scalings of the metric

Let $\overline L= (L, h)$ be a hermitian $C^ \infty$ line bundle on an arithmetic variety $X\to\operatorname{Spec }\mathbb Z$ (I am reasoning in terms of higher Arakelov geometry, like in Gillet & ...
• 619
149 views

### Is the homotopy limit of derived schemes along affine maps a derived scheme?

The title question is true in the setting of ordinary limits and ordinary schemes; that is, given an inverse limit of schemes along affine maps, the limit still lives in the category of schemes. I'd ...
• 271
185 views

### Terminology in log geometry

A log scheme consists of a scheme $X$, a sheaf of monoids $M_X$ on $X$, and a map $\alpha:M_X\to\mathcal O_X$ with the property that $\alpha^{-1}(\mathcal O_X^\times)\to\mathcal O_X^\times$ is an ...
• 17.7k
140 views

### For a family of short exact sequences of coherent sheaves, can we define the splitting subscheme?

This question has been asked in SE. Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a projective scheme over $k$. We can talk about short exact sequences of coherent ...
• 399
68 views

### General fiber and the symmetric product of an ample hypersurface

Let $Sym^m(X)$ be the $m$th symmetric product of a smooth projective variety $X$, $n=\dim(X)$, $Y_1$ an ample hypersurface of $X$, and $CH_0(X)_{hom}$ the Chow groups of $0$-cycles of degree $0$....
• 439
1 vote
156 views

### Meaning of torsion points in a Roitman's theorem

I am having some problems to understand the meaning of the following theorem due to Roitmann. I found this theorem in Voisin's book: Hodge Theory and Complex Algebraic Geometry, Volume II, page ...
• 439
93 views

• 619
1 vote
159 views

• 5,333
88 views

### Closed algebraic subset dominating a curve

In the book "C. Voisin. Hodge Theory and Complex Algebraic Geometry. Volume II. Cambridge studies in advanced mathematics 76 (2002)" page 228 says: Let $X$ be a smooth projective variety. If ...
• 439
1 vote
118 views

### integral subschems of geometrically integeral schemes

Let $X$ be a geometrically integral surface over a field $k$ and let $C$ be an integral curve on $X$. I want to know whether $C$ is geometrically irreducible over $k$ or not.
• 19
1 vote
112 views

### Help to use Statistics and algebra books for community [closed]

My father has 2000 statistics and higher algebra books (schaum series etc). Need to use these for community since he passed away (India) kindly guide me I just need to know if we can donate these ...
380 views

### Extending functors between K-algebras to schemes

Assume we have $K$ and $L$ (comm.) rings, and we have a functor $F$ from the category of $K$-Algebras to the category of $L$-Algebras (I work only with commutative rings). What conditions need to ...
1 vote
368 views

### Intuition behind formal neighborhood and local ring and formal power series

In The Geometry of Schemes by David Eisenbud and Joe Harris, on page 57, there is an explanation on "node" of a plane curve. The book says that, a curve $X\subseteq \mathbb A_{\mathbb C}^2$ ...
• 1,449
231 views

### Flatness of affine cone due to semicontinuity theorem

I would like to clarify an important aspect from the discussion in this question. The OP discussed an obstacle to solve part (c) from Exercise 9.5 from Hartshorne's Algebraic Geometry Chap. III page ...
• 903
1 vote