Questions tagged [ag.algebraic-geometry]

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

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5 votes
1 answer
191 views

Waring rank of monomials, and how it depends on the ground field

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

Calculation of solid angle for rectangle in 6DOF [closed]

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

Deformation theory of stacks and the tangent complex

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

On the stability of having a normal formal model under finite extensions of the base field

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 views

When the sheaf of principal parts is reflexive?

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 views

(Non-)algebraic groups: regularity of multiplication does not imply regularity of Inversion?

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

topological Euler characteristic of canonical divisor

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 views

Reference request: infinity categories for the commutive algebraist/algebraic geometer

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 $\...
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1 vote
1 answer
180 views

One-dimensional family of complex algebraic K3 surfaces

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 ...
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1 vote
1 answer
83 views

For an automorphism of a flat family of sheaves, is there a subscheme of the base where the automorphism is identity?

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

How to compute the exceptional divisor of this blow-up

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 ...
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2 votes
0 answers
125 views

Pushout homotopy squares in motivic homotopy theory

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 ...
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2 votes
1 answer
192 views

Is $\Gamma(U,\mathscr{O}_{X})$ a finitely generated $\Gamma(X,\mathscr{O}_{X})$-algebra?

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

For what properties $ (\mathcal{P}) $ (if any) does $ (\mathcal{P}) $ + analytic isomorphism imply birationality?

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)) $ ...
  • 377
7 votes
1 answer
368 views

Reconstruct a variety from the category of locally free sheaves

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 ...
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8 votes
0 answers
280 views

Intuition for de Rham comparison theorem in $p$-adic Hodge theory

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

Equivariant maps are closed?

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 ...
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1 vote
0 answers
124 views

Closedness of surjective map of affine spaces

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. ...
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4 votes
0 answers
144 views

Numerical Donaldson-Thomas Invariants

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

Non-trivial example of a variety with an action of a unipotent group?

$ \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

Surjective maps of affine spaces are closed

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

Smooth surjective morphism to integral scheme

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 ...
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2 votes
1 answer
77 views

What is the minimal $ n $ such that the Cox ring of the blow up of a simplicial, $ r $-dimensional, toric variety at $ n $ points in g.p. is not f.g.?

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 ...
  • 377
2 votes
0 answers
62 views

Does there exists a "local slice" for an action $ \widehat{\mathbb{G}_{a}} $ on $ \operatorname{Spf}(\widehat{A}) $ (char zero)?

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)...
  • 377
4 votes
1 answer
172 views

Example of a line bundle not admitting a $\operatorname{PGL}(n+1)$-linearization in Mumford's GIT

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

Invariance of numerical class of a curve in Higgs-Grassmann schemes

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

Does analytic isomorphism imply local isomorphism?

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 ...
  • 377
2 votes
0 answers
102 views

Upper bound for the torsion subgroup of an elliptic curve over arbitrary number fields

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

Polynomiality of the equivariant Euler characteristic of a sheaf tensored with a standard line bundle on the flag variety

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

Is $U\subseteq X^{s}(L)$?

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 $...
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2 votes
1 answer
486 views

Proposition 1.5 in Mumford's Geometric Invariant Theory

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

Discriminant of a polynomial form on a vector bundle

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 ...
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3 votes
1 answer
168 views

Resolution of conical singularities have even-only cohomology?

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 ...
  • 1,519
11 votes
1 answer
276 views

Why are these graphs coming from 9-dimensional alternating trilinear forms so symmetric?

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

Frobenius pushforward of an equivariant tautological bundle on the flag variety

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

show that the ring k{{T}} is local [migrated]

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 ...
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-1 votes
0 answers
208 views

What is the obstruction to uniruledness being uninteresting?

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 ...
  • 377
2 votes
1 answer
130 views

Could certain closed covering determine a coherent sheaf?

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 ...
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3 votes
1 answer
93 views

Coordinate principal bundle over a curve

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

Non-Kähler pseudo-Kähler manifolds

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

Existence of a minimal ideal with a specific property

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

Homogeneous components of Cox RIngs

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

What is the kernel of the differential of the orbit-stabilizer map for nonsmooth stabilizers?

$\newcommand{\Lie}{\operatorname{Lie}}$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 ...
  • 161
11 votes
2 answers
484 views

A quadratic $O(N)$ invariant equation for 4-index tensors

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

Semi-continuity of the Picard number

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 $s_0\in S$. ...
  • 107
1 vote
0 answers
242 views

Corollary 1.6 in Mumford's Geometric Invariant Theory

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}\...
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1 vote
0 answers
109 views

How to link the rank of Elliptic Surfaces and Elliptic Curves over function fields?

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(...
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2 votes
0 answers
110 views

Flatness and locally freeness

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

Is the quotient $(\mathcal{O}_X\otimes V)/\mathcal{F}$ torsion-free?

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 $\...
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2 votes
1 answer
96 views

Polystable vector bundle contains a prescribed line bundle as a line subbundle

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}^...
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