# Questions tagged [ag.algebraic-geometry]

Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

20,024
questions

5
votes

1
answer

191
views

The Waring rank of a degree-$d$ homogeneous polynomial $p$ is the least integer $r$ such that you can write $p$ as a linear combination of $r$ $d$-th powers of linear forms $\{\ell_k\}$:
$$
p = \sum_{...

0
votes

1
answer

69
views

I am an undergrad trying to understand and use solid angle calculations:
I have a point source in R3 space (x_source, y_source, z_source) and a rectangle with given center (x_center, y_center, ...

1
vote

0
answers

209
views

On a smooth stack X one can construct the two-term tangent complex $T_X \in D(X)$, as in Sam Raskin's notes (https://people.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept22(Dmodstack1).pdf).
I ...

2
votes

0
answers

67
views

Let $K$ be a finite extension of the $p$-adic numbers with valuation ring $\mathcal{R}$ and uniformizer $\pi$. Consider a smooth and connected rigid $K$-variety $X=Sp(A)$ and assume that the affine ...

2
votes

0
answers

107
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Following EGA IV, we can construct the sheaf of principal parts of a sheaf $\mathcal{E}$ over a scheme $X$ by $\mathcal{P}_X^n(\mathcal{E}) = \mathcal{P}_X^n \otimes \mathcal{E}$, where $\otimes$ is ...

3
votes

0
answers

147
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Fact/motivation:
$\DeclareMathOperator{\inv}{inv}\DeclareMathOperator{\GL}{GL}$
If $G$ is a smooth manifold of dim.$n$ and a group, s.t. the multiplication
$$
m \colon G \times G \to G
$$
is smooth, ...

4
votes

1
answer

234
views

Let $X$ be a complex smooth projective variety with trivial topological Euler characteristic $\chi_{\text{top}}(X)=0$. We assume that $D$ is a smooth irreducible divisor in the linear system $|K_X|$ ...

8
votes

0
answers

404
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In a survey article Algebraic geometry in mixed characteristic, B. Bhatt writes
For instance, given a commutative ring $R$ with a finitely generated ideal $I$,
the assignment carrying $R$ to the $\...

1
vote

1
answer

180
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Let $X$ be an algebraic complex K3 surface, we know that $X$ is deformation equivalent to a smooth quartic surface or more generally a K3 surface with Picard number $1$ (a very general K3 surface in ...

1
vote

1
answer

83
views

Let $p:X\to S$ be a morphism of schemes. Let $\mathcal F$ be an $\mathcal O_X$-modules. Assume that:
$\mathcal F$ is quasi-coherent of finite type;
$\mathcal F$ is flat over $S$;
the support of $\...

0
votes

0
answers

92
views

Suppose that $k$ is a field and $R$ is the ring $k[x,xy,xy^2,xy^3]$.Let $I$ be the maximal ideal of $R$ generated by $x,xy,xy^2,xy^3$.Let $E$ be the exceptional divisor of the blow-up of Spec$R$ along ...

2
votes

0
answers

125
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I am using Vladimir Voevodsky's notes on motivic homotopy theory and I am having trouble understanding the proofs of Corollary 2.20 and Lemma 2.21. Both of these deal with homotopy pushout squares and ...

2
votes

1
answer

192
views

Let $X$ be a scheme of finite type over $\mathrm{Spec}(A)$, where $A$ is a commutative ring. Let $U\subset X$ be any open subset, is $\Gamma(U,\mathscr{O}_{X})$ a finitely generated $\Gamma(X,\mathscr{...

1
vote

0
answers

119
views

In general if $ \mathfrak{p} \in \operatorname{Spec}(A) $ and $ \mathfrak{q} \in \operatorname{Spec}(B) $ are two (not necessarily closed) points such that $ (\mathfrak{p}, \operatorname{Spec}(A)) $ ...

7
votes

1
answer

368
views

We know that the category of the (quasi-)coherent sheaves on a smooth projective variety $X$ determine the variety (aka. Gabriel–Rosenberg reconstruction theorem) and the derived category of coherent ...

8
votes

0
answers

280
views

The de Rham comparison theorem from $p$-adic Hodge theory compares the etale cohomology of a variety with the de Rham cohomology of that variety. It says the following:
Let $K/\mathbf{Q}_p$ be a ...

0
votes

0
answers

181
views

Let $X= \mathbb{A}_K^2 \setminus \{0\}$ with a $K^*$ action given by $c.(x, y)= (cx, cy)$ and $Y= \mathbb{A}_K^2 \setminus \{0\}$ with $K^*$ action given by $c.(x, y) =( c^{d_1}x, c^{d_2}y)$ for some ...

1
vote

0
answers

124
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Let $f:\mathbb{A}^2 \to \mathbb{A}^2 $ be defined by $f(x, y)= (p_1(x, y), p_2(x, y))$, where $p_1, p_2$ are homogenious polynomial of degree $d_1$ and $d_2$ respectively such that $f$ is surjective. ...

4
votes

0
answers

144
views

I happened to read some things about 'numerical Donaldson-Thomas invariants' and the 'integrality conjecture' (by Kontsevich and Soibelman, iirc) and I was hoping someone would shed light on what ...

4
votes

0
answers

161
views

$
\renewcommand{\C}{{\mathbb C}}
\renewcommand{\R}{{\mathbb R}}
$
In the preprint Taking quotient by a unipotent group induces a homotopy equivalence
we proved the following result:
Theorem.
Let $U$ ...

2
votes

1
answer

189
views

Let $f: \mathbb{A}^n \to \mathbb{A}^n$ be a quasi-finite surjective morphism.
Question: Is $f$ closed ?

1
vote

0
answers

107
views

Suppose that $f : X \rightarrow Y$ is a smooth (or even étale) surjective morphism over a field $k$ to a scheme $Y$ of finite type over $k$.
I want to show that $X$ is locally integral, i.e. (in the ...

2
votes

1
answer

77
views

In Shigeru Mukai's paper "Counterexample to Hilbert's 14th Problem for the 3-dimensional Additive Group," Mukai proved that if $ \frac{1}{r+1}+\frac{1}{n-r-1} \le \frac{1}{2} $, then the ...

2
votes

0
answers

62
views

Every action $ \beta $ of $ \mathbb{G}_{a} $ on a variety $ \operatorname{Spec}(A) $ over a field of characteristic zero is obtained from a locally nilpotent derivation $ \delta $ via $ f(t_{0} \ast x)...

4
votes

1
answer

172
views

$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Proj{Proj}\DeclareMathOperator\Pic{Pic}$I have a question about an example for a line bundle not admitting a
$G$-linearization from Mumford's GIT, ...

1
vote

0
answers

77
views

Premise
Let $X$ be a projective variety of dimension $n\geq1$ over an algebraically closed field of characteristic $0$.
A Higgs sheaf $\mathfrak{E}$ is a pair $(E,\varphi)$ where $E$ is a $\mathcal{...

1
vote

0
answers

154
views

If $ \mathfrak{p} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(A) $, and $ \mathfrak{q} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(B) $ such ...

2
votes

0
answers

102
views

Let $K$ be a finite extension of $\mathbb{Q}$ of degree $d$ and let $E(K)$ be an elliptic curve over the field $K$ with coefficients in $K$. Let us fix $d$ and vary over all the possible $K$, in turn ...

1
vote

0
answers

53
views

Let ${\mathcal B}=Fl(V)$ be the variety of complete flags in an $(m+2n)$-dimensional vector space $V$ over $\mathbb C$. Then we have the standard line bundles $\mathcal{O}(\lambda)$ on $\mathcal B$ ...

1
vote

0
answers

87
views

Let $X$ be an irreducible affine variety over an algebraically closed field $\mathbb{K}$ of characteristics $0$. Let $G$ be a connected, linearly reductive, affine algebraic group acting regularly on $...

2
votes

1
answer

486
views

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\pr{pr}$I have some problems to understand the proof of Proposition 1.5 from Mumford's ...

0
votes

0
answers

81
views

Let $\mathcal E$ be a rank $2$ vector bundle on a smooth (complex) variety $X$ and $\mathcal L$ a line bundle.
Consider $q\in H^0(X,{\rm Sym}^d\mathcal E\otimes \mathcal L)$ ($d>2$) a polynomial ...

3
votes

1
answer

168
views

Considering a quotient singularity $\mathbb{C}^n/G,$ its crepant resolution $Y$ (i.e. having $c_1(Y)=0$) has rational cohomology supported in even degrees only. This holds for many other resolutions ...

11
votes

1
answer

276
views

Let $\phi(x,y,z)$ be an alternating trilinear form on a space $V$ over a field $K$.
Let $u \in \mathbb{P}(V)$ be a projective point over $V$, then we say that the rank of $u$ is equal to the rank of ...

4
votes

1
answer

249
views

Let $m,n$ be nonnegative integers. Let $k$ be a field of positive characteristic. Let the vector space $V$ over the field $k$ have basis $e_1,...,e_{m+n}, f_1,..., f_n$. Let $k^*$ act on $V$ by $t \...

-1
votes

0
answers

29
views

I have to prove that k[[T]] is local and I've use this argument: we have this serie $f(T)$=$a_0 + a_1T+ a_2T^2+...$ with $a_0 \not = 0$.
We can find $g(T)$ such that $f(T)·g(T)=1$
So we have some ...

-1
votes

0
answers

208
views

An $ n $-dimensional projective variety $ Z $ over a field $ k $ (assume characteristic zero) is uniruled if there is an $ n-1 $-dimensional variety $ Y $ and a dominant, generically finite, rational ...

2
votes

1
answer

130
views

We know that a coherent sheaf on a scheme is determined by its restriction on certain open coverings (satisfying compatibility condition). Now I wonder how about a closed covering. To do so I started ...

3
votes

1
answer

93
views

I am trying to understand more about geometric interpretation of vertex algebras following "Vertex Algebras and Algebraic Curves" by Ben-Zvi and Frenkel, but I am in trouble with the ...

11
votes

2
answers

573
views

A pseudo-Kähler manifold is a complex manifold $(X, I)$ endowed with a non-degenerate closed $(1, 1)$-form $\omega$. In that case, the symmetric tensor $g(\cdot, \cdot) = \omega(\cdot, I \cdot)$ is a ...

1
vote

0
answers

38
views

Suppose that $R$ is a super-commutative ring (i.e. it is a unital $\mathbb{Z}_2$-graded ring satisfying $xy=(-1)^{|x|\cdot |y|}yx$ where $|x|$ denotes the grading degree of a homogeneous element $x\in ...

1
vote

1
answer

78
views

Let $X$ be an irreducible smooth projective variety over a field $k$ (algebraically closed and of characteristic zero if needed). Let $U \subseteq X$ an affine open such that $O_X(U)$ is factorial and ...

1
vote

0
answers

30
views

$\newcommand{\Lie}{\operatorname{Lie}}$Let $G$ be a smooth linear algebraic variety over perfect field $k$, acting on a separated variety $X$, and for $x \in X(k)$ write $G_x$ for the scheme-theoretic ...

11
votes

2
answers

484
views

Consider an $O(N)$ invariant quadratic equation
$$
T_{ijkl}= T_{ijmn}T_{klmn}+ T_{ikmn}T_{jlmn}+ T_{ilmn}T_{jkmn},
$$
where $T_{ijkl}$ is a real, totally symmetric 4-tensor, and the indices run from 1 ...

2
votes

0
answers

142
views

Let $f:X\rightarrow S$ be a family of smooth projective varieties. For $s\in S$ set $X_s := f^{-1}(s)$, and let $\rho(X_{s})$ be the Picard number of the fiber over $s\in S$. Fix a point $s_0\in S$.
...

1
vote

0
answers

242
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I do not understand fully the proof of Corollary 1.6 from Mumford's Geometric Invariant Theory (§3: Linearization of an invertible sheaf, p 35):
Corollary 1.6
$\DeclareMathOperator\Spec{Spec}\...

1
vote

0
answers

109
views

I have been investigating certain elliptic surfaces for my research, and when giving a presentation, I was asked why when given an elliptic surface $E$, say given in Weierstrass form
$$E\colon y^2+a_1(...

2
votes

0
answers

110
views

Let $X$, $S$ be integral quasi-projective schemes (over $\Bbb C$). Let $\mathcal F$ be a coherent sheaf on $X\times S$, flat on $S$. Suppose that $x\in X$, $s\in S$ are closed points, and ${\mathcal F}...

1
vote

1
answer

160
views

Consider a smooth projective variety $X$ and an exact sequence
$$0\rightarrow\mathcal{H}\rightarrow\mathcal{O}_X\otimes V\rightarrow\mathcal{G}\rightarrow0$$
where $V:=H^0(X,\mathcal{G})$ and $\...

2
votes

1
answer

96
views

Let $X$ be a compact Riemann surface of genus $g \geq 1$, and let L be a line bundle over $X$ with $-g < \deg L \leq -\frac{1}{2}g$. Can we always find a flat line bundle $J \in \operatorname{Pic}^...