Questions tagged [algebraic-geometry]
The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.
27,393
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Bijective map between two definition for extension for sheaf on base
Let $F$ be sheaf on base $\mathscr B =${$B_i,i \in I$}, there are two equivalent definitions for the extention for $F$, denoted by $\mathscr F $:
Def1:$\mathscr F(U) $ := {$(f_p \in F_p)_{p \in U}:$ ...
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30
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Proving the Relative Nullstellensatz
Let $X \subseteq \mathbb{A}^n$ be an affine subvariety and let $A(X)$ be the corresponding coordinate ring. For $Y \subseteq X$ and $S \subseteq A(X)$ we define the following relative analogues to the ...
- 3,562
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Inverse and direct limits and cohomology
Suppose $X_n$ is a tower (an inverse system), indexed by the positive integers, of smooth projective complex varieties, and call $X = \varprojlim_n X_n$ as schemes over $\mathbf{C}$.
Fix $\ell$ a ...
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Dominant morphisms between projective varieties
Suppose $f : X\to Y$ is a finite surjective morphism of projective integral complex varieties, and let $g : X'\to X$ and $h : Y'\to Y$ be surjective birational morphisms from smooth projective ...
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What is the relation between an affine variety and an affine space?
I was learning about algebraic varieties and the wikipedia page presents affine varieties as the "conceptually easiest" type. Having read about affine spaces and affine varieties, I am ...
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Figure interpretation: $p$ in the affine plane is replaced by a projective line in the blow up.
In my text book the point in the affine plane is supposed to be replaced with "all the lines through $p$, a copy of $\mathbb{P}^{n-1}$." When I examine the figure of the blowup, there is one ...
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height of regular sequence modulo minimal prime
Liu Qing's book Algebraic Geometry and Arithmetic Curves has the following statement on 6.3.11:
Let $Y$ be a locally Noetherian scheme and $i \colon X \to Y$ a regular immersion of codimension $n$. ...
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When is the tangent space of a fiber product the fiber product of the tangent spaces?
If $X,Y,Z$ are schemes locally of finite type over an algebraically closed field of characteristic 0, and $X\to Z$ and $Y\to Z$ are morphisms, then is it true that $T_{(x,y)}(X\times_Z Y)\simeq T_x(X)\...
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Base gluability axiom
In Vakil's "the rising sea" book, the base gluability axiom is defined as:
"(Let $F$ be a sheaf on base {$B_i$}) If $B = \cup B_i$, and we have $f_i \in F(B_i)$ such that $f_i$ agrees ...
- 13
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The Coordinate ring $A(X)$ is a field if and only if $X$ is a singleton
Let $X \subseteq \mathbb{A}^n$ be an affine variety. Show that the coordinate ring $A(X)$ is a field if and only if $X$ is a singleton.
Remark: This is exercise 1.20 from Gathmann's Notes "...
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Show that the image of an irreducible affine variety is irreducible [duplicate]
I would really appreciate some help on this:
Definition: Let $k$ be a field. An affine variety is a space with functions $(X, O_X)$ i.e. a topological space equipped with a sheaf of $k-algebras$, ...
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What Irreducible cubic curve and Siegel's theorem
Let $c(x, y)$ be a cubic polynomial with rational coefficients which is irreducible over $\mathbb{Q}$. I'm trying to understand when the equation $c(x) = 0$ has infinitely many integral points. If $c$ ...
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Differentiable way of detecting characteristic classes of odd (real) dimension vector bundles?
I learned that Chern classes work for complex vector bundles (corresponding to a real bundle of even dimensions). Equivalently, Pontryagin classes work for even dimensions.
Is there a characteristic ...
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Show that divisor in $\mathbb P^2\times \mathbb P^2$ is very ample.
Let $H$ denote a quadric hypersurface in $\mathbb P^2\times \mathbb P^2$. In the Chow ring of $\mathbb P^2\times \mathbb P^2$, we have $H\equiv 2H_1+2H_2$, where $H_1,H_2$ are the classes of linear ...
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Autoequivalences of $\operatorname{Coh}(X)$
Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic zero.
Is there a description of $\operatorname{Aut}(\operatorname{Coh}(X))$, i.e. the autoequivalences ...
- 1,740
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proof on regular functions on complete connected varieties is constant
The complete reference is Cor 7.24 of this notes: https://www.mathematik.uni-kl.de/~gathmann/de/alggeom.php
We want to prove the statement "Let $X$ be a connected complete variety, then $\mathcal{...
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The difference of the vector bundle's definitions between the Hartshorne and the Stacks Project
I currently learn scheme theory by self-learning racking my brains. In that time, a difference give me a headache: The difference of the vector bundle's definitions between Exercise II.5.18. from the ...
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Question on smooth affine curves over finite fields and Hasse-Weil
I don't have a strong background in algebraic geometry, so my question could seems trivial, but I would like to know more about it. So if you have a book to recomend, thanks in advance!
Suppose you ...
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Elimination property on scheme morphisms locally of finite type
I was trying to prove the following scheme morphism exercise. If $f: X \rightarrow Y, g: Y \rightarrow Z $ are such that $g \circ f $ is locally of finite type then $f$ is locally of finite type. I ...
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Motivations for smooth morphisms, unratified morphisms and étale morphisms
I have learnt basic definitions and some properties by reading stacks project which is good reference.
However, in my stupid view, I can not understand why we define them in this way.
I have read two ...
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Elements in projective space
Consider the projective line for simplicity. I was wondering was it means to look at the stalk at $0$. What is $0$ even referring to? The ideal generated by $0$ in one of the copies of the affine line?...
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Dense algebraic curves
Let's assume we are given $\mathbb{R}^2$ with the standard topology (unions of open balls).
We call a subset $A \subset \mathbb{R}^2$ dense, if $\forall U \underset{open}{\subset} \mathbb{R}^2 \ \...
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If special fiber of $S$ is smooth, can we say $S$ itself is smooth?
Let $X$ be an equation given by $Y^2Z=X^3-XZ^2$ in $\Bbb{P}^2$.
Let $A$ be a DVR with maximal ideal $P$ and $S$ be a scheme over $A$ defined by $X$, i.e. $S=Spec A[X,Y,X]/(Y^2Z-X^3-XZ^2)$.
If special ...
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characteristic classes in odd dimensions? [closed]
So Stiefel-Whitney, Pontryagin, Chern classes are all for even dimensions. There seem to be no characteristic classes for odd dimensions. Does that mean in odd dimensions everything is trivial?
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Lifting Primary Decomposition Over DVRs From Their Field of Fractions
Suppose we have an ideal $I = (f_1,...,f_m) \subset \mathcal{O}[x_1,...,x_n]$ where $\mathcal{O}$ is a DVR with uniformizer $r$. Let $K := \text{Frac}(\mathcal{O})$. Let $I_K$ be the ideal of $K[x_1,.....
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Ideal of definition of local ring
Definition. Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$. An ideal $I$ is called an ideal of definition of $R$ if $\mathfrak{m}^n \subset I \subset \mathfrak{m}$ for some $n\...
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Hartshorne Exercise II 6.11 (c)
Exercise II 6.11:
Let $X$ be a nonsingular curve over an algebraically closed field $k$.
(c) If ${\mathscr{F}}$ is any coherent sheaf of rank $r$ (means that its stalk at the generic point has ...
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Type A quivers and flag varieties from Kirilllov
I'm trying to understand section 10.7 of Kirillov's "Quiver representations and quiver varieties", which shares its title with this question. We let Q be a type A quiver of length $\ell$. We ...
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Locally free resolution of coherent sheaves on nonsingular curves
This question is from Exercise II 6.11 of Hartshorne.
Let $X$ be a nonsingular curve over an algebraically closed field $k$. For any coherent sheaf $\mathcal{F}$ on $X$, show that there exist ...
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Can we infer Betti numbers of a family degenerating to a variety from the variety?
I'm noticing I know positive results to the following question in a number of special cases, but I don't know the general situation.
Let $S$ be a complete trait, to make sure I'm not misspeaking ...
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The degree of a morphism of schemes
Let $X$ and $Y$ be schemes over a field $k$, and let $\phi :X \to Y$ be a morphism of schemes.
What is the most general situation where one can define the degree of $f$? Is $X$ and $Y$ being ...
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A question about the number of degenerate cubic curves of a pencil
I have read a post about counting degenerate cubic curves in the pencil of two non-degenerate cubic curves.
Count degenerate cubic curves in the pencil of two non-degenerate cubic curves There is an ...
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Rees algebras: why do we have this bijection on minimal primes?
I'm learning a little bit about Rees algebras lately, and I'm reading from Swanson and Huneke's Integral Closure of Ideals, Rings, and Modules (available here on one of the author's websites) and ...
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$S^1$ action on coherent sheaves
Given a superpotential $W: X \rightarrow \mathbb{C}^\ast$, it is said that $W$ defines canonically an $S^1$ group action on coherent sheaves of $X$. How is this the case?
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How to get an action of topological fundamental group on the singular cohomology of a fiber?
Suppose $f:X\to Y$ is a proper smooth morphism of $\mathbb C$-varieties, and $y\in Y$ is a point. I want to get an action of $\pi_1(y,Y)$ (topological fundamental group) on $H^i_{sing}(X_y,\mathbb C)$....
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Vector bundles on curves and Adeles
I have heard of the following 'well-known' result attributed to Weil, that rank $ n $ vector bundles on a curves over a field can be obtained as double quotients of $ \operatorname{GL}_n $ of the ...
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Discrepancy in effective Cartier divisors
In The Stacks project effective Cartier divisors are defined as closed subschemes whose ideal sheaf is invertible.
In Hartshorne, page 145, an effective Cartier divisor is represented by $\{(U_i,f_i)\}...
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Curves homeomorphic under Zariski topology
Prove any two curves over some field $k$ are homeomorphic, where $k$ might not be algebraically closed. Curves are defined to be varieties (integral separated scheme of finite type) of dimension $1$.
...
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Geometric interpretation of Minimal Prime Ideals
I'm doing Gathmann's commutative algebra notes Exercise 2.23:
Let $I$ be an ideal in a ring $R$ with $I\neq R$. We say that a prime ideal $P\unlhd R$ is minimal over $I$ if $I ⊂ P$ and there is no ...
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Classify U(1) bundle over $\mathbf{P}^3$, and its topological invariants
I am interested in knowing the classification of the U(1) bundle over the complex projective space $\mathbf{P}^3$. This is effectively a U(1) bundle over the 6-manifold $M^6$.
What are the possible ...
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Dimensions of cohomology of ideal sheaf
This question follows a previous question Sheaf morphism from closed subscheme is a closed immersion, it's just another part so I'll recall everything.
For $K=\bar{K}$ a field consider $X=\mathbb P^...
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Mayer Vietoris sequence and pair long exact sequence [closed]
My question is
Do Mayer Vietoris exact sequence and excision theorem holds for every cohomology and homology？ If no， does have counterexample? For example, for galois cohomology , I never find any ...
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Reference request: Intersection number using derived tensor product
I just learned there is a version of definition of intersection number using derived tensor product to avoid the moving lemma. I only know intersection theory as presented in Fulton's book. Does ...
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lines passing through two points in $\mathbb{P}^n$
The reference is the Example 7.5. b) from ag notes by Gathmann. The paragraph explains the idea of projection from a point. He claims that the unique line passing through $a=(1:0:\cdots:0)\in\mathbb{P}...
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Is there $f: \mathcal{O} \oplus \mathcal{O} \to \mathcal{O}(1) \oplus \mathcal{O}(1)$ with cokernel $\mathcal{K}_P$ for some $P$ of degree 2?
Let $X$ be the projective line over a field $k$ of positive characteristic. I am trying to prove that don't exist an exact sequence of coherent sheaves over $X$
$$0 \to \mathcal{O} \oplus \mathcal{O} \...
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universal coefficient theorem and kunneth theorem for all cohomology theorey in math [closed]
We know universal coefficient theorem and kunneth theorem for cohomology or homology of CW complex or manifold. However, I wonder whether they also holds for sheaf cohomology or etale cohomology or ...
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Show that $V_Y(J)=V(\pi^{-1}(J))$ - Relative Nullstellensatz
Let $Y \subset \mathbb{A}^n_k$ and define the coordinate ring A(y) by the quotient
$$
\pi : k[x_1,\ldots,x_n] \to A(y):=k[x_1,\ldots,x_n]/I(Y)
$$
Show that $V_Y(J)=V(\pi^{-1}(J))$ for every ideal $J$ ...
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Mazur's theorem for general number fields
Mazur's theorem completely classifies the possibilities of $E_{tors}(\mathbb{Q})$ for an elliptic curve $E/\mathbb{Q}$, including the fact that only finitely many groups occur. What happens with $E_{...
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Rational points on fiber product
I am trying to figure how rational points in fiber products look like. Let $K$ be a field, $A=K[x_1, \ldots, x_n], B=K[y_1, \ldots, y_m]$. Rational points in SpecA and SpecB look like $(x_1-a_1, \...
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Why do we assume that the residue field extension is separable in the definition of unramified
Neukrich defines a finite extension $L / K$ of henselian fields to be unramified if the extension of residue fields $\lambda / \kappa$ is separable and $[L : K] = [\lambda : \kappa]$ (emphasis added).
...
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