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.

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19 views

Request for reference for currents

Do you know an introductory book (end of undergraduate/beginning of graduate) about currents in the sense of de Rham? Or even better, a video lecture on that topic? I didn't find it on youtube. Any ...
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29 views

Graduation of Clifford algebra

Given a vector space $V$ and a quadratic form $q$ on it, we know that the Clifford algebra $Cl(V,q)$ is a graded algebra, which is a property inherited by the Tensor algebra $T(V)$. Unless in the ...
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18 views

First cohomology group of a Lie algebra is isomorphic to $L/[L,L]$?

Let $L$ a Lie algebra on a field of characteristic zero with finite dimension greater than one. I have to show that $H^1(L) \cong L/[L,L]$. Now I write my proof of this statement. First of all I ...
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1answer
39 views

Why is $m \geq n$ obvious in Vakil's proof of Noether Normalization (11.2.4)?

This is a very specific part of a classic proof so I had trouble finding duplicates; I apologize if it is one. We have $A$ a finitely generated $k$-algebra which is an integral domain. As such, we ...
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1answer
18 views

Specializing Weak Factorization to Birational morphism

The Weak Factorization Theorem tells us that birational map of varieties over field of perfect characteristic which has resolution of singularities can be factored into blow-ups and blow-downs. My ...
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Line bundles on complete flag varieties independent of isogeny class

Let $G$ be a semisimple connected linear algebraic group over an algebraically closed field $k$. If $G^{sc}$ is the simply-connected cover of $G$ (i.e., the semisimple connected simply-connected ...
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1answer
34 views

Surjections/injections seen through quantifier adjoints

Any map of sets $A \xrightarrow{f} B$ induces a functor of posets $Sub(B) \xrightarrow{f^{-1}} Sub(A)$. It is known to have left and right adjoints $\exists_f$ and $\forall_f$ (defined as $\exists_f: ...
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24 views

Is the quasi-projective scheme morphism separated and finite type?

Is the quasi-projective scheme morphism separated and finite type? In Hartshorne's book, it is shown when it is noetherian, but I was wondering if this assumption is necessary.
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29 views

Constructing dual variety using duals of singular points

let $X\subset\mathbb P\mathbb C^n$ be a closed, irreducible (reduced) variety. Then the dual variety $X^*$ can be defined as the projection of the closure of the conormal variety $$ \mathrm{Co}(X)=\{(...
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1answer
50 views

Invertible functions on a proper variety

It seems well-known that for a smooth proper geometrically integral variety $X$ over a number field $k$, the invertible functions on $\bar{X} := X \times _k \bar{k}$ are the nonzero constants, i.e., $\...
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1answer
49 views

How does a (finite) group act on a(n affine) scheme?

I understand what it means for a group scheme to act on a scheme, but I don't understand what it means for a finite group to act on a scheme. I am sure it doesn't mean that the group acts on the ...
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43 views

Injectivity of Keller maps

Let $M: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$, $(x,y) \mapsto (p,q)$, with $p,q \in \mathbb{C}[x,y]$ satisfying $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$. Such a polynomial map is ...
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Is the set of coefficients of $n+1$ homogeneous polynomials that have unique root $p_{0}$ open?

Define the set $U=\{(a_{0},\ldots,a_{n}): F_{a_{0}}(p_{0})=\ldots=F_{a_{n}}(p_{0})=0, p_{0} \in \mathbb{P}^{n}\}$, where $F_{a_{i}}$ is homogeneous polynomial of deegre $d_{i}>0$ and $a_{i}$ is the ...
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1answer
40 views

Given $h \in k[x]$, $F(T) \in k[x][T]$ a multiple of $T$, can we find $g \in k[x]$ not a multiple of $h$, such that $F(g)$ is a multiple of $h$?

Let $k$ be a field of characteristic zero, for example $k=\mathbb{C}$. Let $F(T) \in k[x][T]$ a multiple ot $T$, namely, $F(T)=f_nT^n+f_{n-1}T^{n-1}+\cdots+f_2T^{2}+f_1T$, where $f_n,\ldots,f_1 \in k[...
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Local Models for shifted symplectic structures

I have a problem to get the signs right in the calculation in the proof of Prop 1.21 of the paper https://arxiv.org/pdf/1111.3209.pdf on shifted symplectic structures. The authors obtain a ètale local ...
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1answer
145 views

Can blow-up of a surface be a product of two curves?

Is there any smooth projective surface $S$ over $k=\bar{k}$, such that the blow up $\tilde{S}$ along some point $x\in S$ can be written as $\tilde{S}=C_1\times C_2$ for two curves $C_i$?
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Is pullback of an injective sheaf to an infinite intersection of opens still injective?

It is well-known that if you have an injective sheaf of abelian groups on a scheme, restriction of this sheaf to any open sub-scheme is also going to be an injective sheaf. The idea is that pullback ...
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43 views

Local etale cohomology for sheaves of abelian groups.

I have seen local cohomology (cohomology supported on a closed subspace) in different contexts, like for topological spaces and for quasi-coherent sheaves on a scheme. I was wondering whether the same ...
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1answer
34 views

Zero locus of a homogeneous ideal.

I did not understand this lemma, could someone explain it to me? Maybe with a concrete example. We use the subscripts "pro" to distinguish the zero sets in affine and the projective setting. ...
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3answers
64 views

Question about not isomorphic mapping between affine varieties

I try to understand the following example: $ F :\mathbb{A}^1 \to \mathbb{A}^2 , t \mapsto (t^2,t^3)$. And let $X=\mathbb{A}^1$, $Y= \operatorname{Im}F=\left \{ (x,y) : F(t)=(x,y) \right \}$ (In fact, $...
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58 views

What does « problèmes nonlinéaires » specifically mean?

In section 1.4 of SGA $4\frac{1}{2}$, there is a brief explanation of why the introduction of the notion of the flat topology is necessary, which says: Dans le cadre des schémas, la topologie de ...
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66 views

Why does Chow's theorem imply that compact complex manifolds embedded in projective space are algebraic?

I am trying to understand when compact complex manifolds are algebraic. In many sources, they prove a certain complex manifold can be embedded into projective space (e.g. with the Kodaira embedding ...
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1answer
63 views

Is there exist an irreducible subvariety dominating the given one?

Let $f$ be a surjective morphism from a projective variety $X$ to a projective variety $Y$ and L a line bundle on $Y$. I want to prove that if the pull back of L is nef, then L is nef. I want use the ...
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25 views

$\mathcal{O}$-connectedness is preserved by flat base change

Let $f:X\rightarrow Y$ be an $\mathcal{O}$-connected morphism i.e. $f_*\mathcal{O}_X\cong\mathcal{O}_Y$ via the natural map, and let $g:Y'\rightarrow Y$ be a flat morphism with the following ...
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1answer
135 views

Projectivity of analytic $\mathbb{P}^1$ bundles

Let $f: X\to Y$ be a smooth analytic $\mathbb{P}^1$-bundle from a complex manifold $X$ to a complex projective manifold $Y$. Is $X$ a projective manifold?
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164 views

Representations of an algebraic group $G$ versus representations of the group $G(k)$

The following question stems from the discussion here. Let $G$ be a group scheme, $G$ acting on a vector space $V$ over a field $k$ is morphism of group valued functors $G \rightarrow GL_V$, where $...
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37 views

On well-definedness of Shimura curve.

Suppose that we have a quaternion algebra $D$ over a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We assume that $D$ splits everywhere at finite places of $K$ and at ...
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31 views

Lefschetz hyperplane theorem for etale homotopy groups

Lefschetz hyperplane theorem : Let $X$ be a smooth projective variety over $\mathbb{C}$ and $H$ be a hyperplane section of $X$ such that $X \setminus H$ is a smooth affine(automatically true in this ...
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1answer
72 views

The locus of $y$ where $B\otimes \mathfrak{m}_y$ is globally generated is open

I got this question when reading Lazarsfeld's book "Positivity in algebraic geometry I". It's example 1.2.9. He claimed the following: "Let $B$ be a globally generated line bundle on a ...
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2answers
78 views

can I use Affine cones for calculating singularities on projective quadrics?

So I wanted to calculate the singularities of a quadric in the projective space. For example if you have the quadric $$ Q=V(X_0^2+X_1^2+...+X_r^2), r<=n $$ Which lives in the projective space $\...
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1answer
64 views

Why is this bundle positive?

Let $E$ be a holomorphic bundle (with rank $r$) over a compact complex manifold $X$. $E^{*}$ will denote its dual. $\mathbb{P}(E^{*})$ will denote the projective bundle associated to the dual of $E$ ...
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1answer
95 views

Algebraic(!!) version of the Riemann existence theorem

In Appendix B to Hartshorne's Algebraic Geometry, Hartshorne claims that one can prove that compact Riemann surfaces are algebraic in the following way. First show there exists a nonconstant ...
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1answer
26 views

Genus of Abelian Varieties

The notes I was reading mentions that if $X$ is an abelian variety of dimension $g$, over a field $k$, then we have the following cohomology: $$H^i(X,\mathcal{O}_X)=k^{g \choose i},\ 0\le i\le g.$$ ...
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1answer
72 views

Formal Definition of closed subscheme

Let $(X,\mathcal{O}_{X})$ be scheme let Z be a closed subset of X ,i have read many definitions of closed subscheme in many books like in Hartshorne , Liu,Qing ,, Ulrich Görtz but i could 't ...
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251 views
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About Theorem 3.4 Hartshorne: detailed proof.

I propose a detailed version of part of the proof of Theorem 3.14 from Hartshorne's book Algebraic Geometry. The questions are inserted from time to time within the proof. Thanks for your patience. ...
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32 views

Isomorphism of abelian varieties [closed]

In [BGK+18] in section 4, Boneh et al. write that: For any choice of ideal classes $\mathfrak{a}_1,\dots,\mathfrak{a}_n,\mathfrak{a}_1',\dots,\mathfrak{a}_n'$ in ${Cl}(\mathcal{O})$, the abelian ...
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1answer
51 views

ring of $p$-adic algebraic integers

Let $K/\mathbb{Q}_p$ be a finite extension, and $\overline{K}$ be some fixed algebraic closure of $K$. Let $\mathcal{O}_{\overline{K}}$ be the ring of all algebraic integers in $\overline{K}$. I know ...
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1answer
57 views

Proving a surjective k-algebra homomorphism to be an isomorphism

I am trying to prove the following statement: Let $f:A\to B$ be a surjective $k$-algebra homomorphism. Let $A$ and $B$ be local Noetherian rings with same finite Krull dimension. Then $f$ is an ...
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1answer
62 views

Question about $\mathbb{A}_\mathbb{Q}^2$

This question comes from Vakil's notes on algebraic geometry: Describe the maximal ideals of $\mathbb{Q}[x,y]$ corresponding to a) $(\sqrt{2},\sqrt{2})$ and $(-\sqrt{2},-\sqrt{2})$ b) $(\sqrt{2},-\...
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1answer
53 views

Line bundles on projective plane [closed]

Suppose $C \subset \mathbb{P}^{2}_{k}$ is a smooth, geometrically irreducible curve defined by degree $d$ polynomial over a field $k$ of characteristic zero. Why is $$ \mathcal{O}_{\mathbb{P}^{2}}(C) \...
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26 views

Question about the degree of finite morphism of curves

I have a question about a definition in Hartshorne chapter II.6. Given a finite morphism $f:X\to Y$ of curves Hartshorne defines the degree of $f$ to be $[K(X):K(Y)]$. I don't understand though why $K(...
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1answer
30 views

Two canonical constructions for a scheme map $\operatorname{Spec} \kappa(x) \to X$

Let $X$ be a scheme and $x \in X$. Then there is a canonical scheme morphism $\operatorname{Spec} \mathcal O_{X,x} \to X$ (Vakil’s Exercise 6.3.J(a)) and the quotient map $\mathcal O_{X,x} \to \kappa(...
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1answer
142 views

Why do we always need the Schwarz lemma when bounding the trace of a Kähler metric?

My undergraduate thesis topic is Kähler geometry. The general direction is something like the Calabi-Yau theorem or more adventurously some singular Calabi-Yau theorem, but this is not certain yet. ...
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1answer
82 views

Chern character of a coherent sheaf over a projective manifold

I am looking for a reference describing the definition/construction of the Chern character of a coherent sheaf over a projective manifold (or variety) using a resolution by vector bundle.
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22 views

Proving the tangent hyperconic of an algebraic hypersurface $V(f)$ is indeed a hyperconic

Suppose $V(f)$ is an algebraic hypersurface in $\mathbb{C}P^n$ and $P=[p]\in V(f)$ a point of the algebraic hypersurface with multiplicity $k$. That is to say, all $k-1$ derivitives of $f$ are zero in ...
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0answers
48 views

The textbook of Abelian varieties included Grothendieck’s semi-stable reduction Theorem

I would like to learn abelian varieties, semi-stable abelian varieties, and the proof of Grothendieck’s semi-stable reduction Theorem. I have briefly learned abstract algebra, category theory, ...
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1answer
61 views

Can a finite locally free morphism of integral affine schemes not be free?

Let $f:X\to Y$ be a finite locally free morphism of integral affine schemes. Is $f$ necessarily free? Is there an easy counterexample to this (eg, a morphism of varieties over an algebraically closed ...
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2answers
178 views

Doubt regarding pushforward followed by pullback of sheaves

The answer might be well-known, but I couldn't find it antwhere. Sorry for my lack of knowledge. Let $\require{AMScd} \begin{CD} Y' @>i'>> X\\ @Vf'VV @VVfV \\ Y @>i>>Z \end{CD} $ be ...
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1answer
18 views

For an affine $k$-scheme of finite type, is the underlying ring a finitely generated $k$-algebra? [duplicate]

If $X$ is an affine $k$-scheme of finite type, say, $X=\operatorname{Spec} A$. Can we deduce $A$ is a finitely generated $k$-algebra? Could you prove it or give a counterexample? Thanks!
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1answer
67 views

Can anyone recommend a good book or a basic book on rational cubic forms?

I am working in the space of skew-symmetric matrices. For $6\times6$ matrices using pfaffian, I think rational cubic forms will help me for further progress. Also, if there is grassmanian, generic ...

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