# Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

14,707 questions

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48 views

### Flatness of the integral closure

Let $R$ be a $p$-torsion free ring which is integrally closed in $R[1/p]$ and let $S$ be a finite etale extension of $R[1/p]$.
Is it true that an integral closure $S^+$of $R$ in $S$ is flat over $...

**2**

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46 views

### Finding a curve through divisor

Let $k$ be a field. Is it true that for any smooth irreducible projective $k$-variety $X$ and a dense open set $U\subset X$, for any zero-cycle on $X$ one can find an irreducible curve containing its ...

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36 views

### On the exponent of a certain matrix $A$ in characteristic $p > 0$

Let $A$ be a square matrix in characteristic $p > 0$ with both column and row having length $(1 + p + \cdots + p^i)$, where $i > 0$.
Suppose that further the $(m,n)$-component $a_{m,n}$ of the ...

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50 views

### Euclidean Geometry [on hold]

In Euclidean Geometry we can create a square with side length of $\sqrt2$. as we know, $\sqrt2$ is a number which has no end. so it is physically impossible to have a square with this length in the ...

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50 views

### Example coordinates for a dodecahedron in x, y, z? [on hold]

Good morning/afternoon/evening all,
I'm working on some high-level Python simulations and wanted an example set of coordinates for the vertices of a regular dodecahedron in terms of x, y and z ...

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230 views

### Confusion on summand of Hochschild homology of D-modules

I've encountered the following confusion.
Start with the category of perfect D-modules on $\mathbb{A} ^{1}$, which I'll denote $D$. We have the object $\mathcal{O}$, which is exceptional in the sense ...

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88 views

### Status of locality of perfectoidness for uniform rings

Let $k$ be a perfectoid field of zero characteristic. Recall that a Tate $k$-algebra is called uniform if the set of power-bounded elements is bounded. Let $(A, A^+)$ be a uniform complete affinoid $k$...

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53 views

### Regular intersecting family

Let $n=2k+1$. When $k=3$, the set of lines of Fano plane is a regular intersecting family consisting of some k-subsets of [n]. Do anyone know such examples for general $k$? Thanks.

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114 views

### Does $\delta$-invariant give a sufficient condition for flatness for plane curve singularities?

Let $\pi:\mathcal{C}\to S$ be a morphism of schemes such that $\mathcal{C} \subset \mathbb{C}^2 \times S$ with the inclusion map commuting with the natural projection to $S$ and for all $s \in S$, $\...

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227 views

### Sheafification map is surjective

This is not a research level problem for sure. But, similar question was asked by some one else $2$ years back on Stack exchange has not received any attention. So, I thought it does not suit there....

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388 views

+500

### Applications of integral p-adic Hodge theory

What are some geometric applications of integral p-adic Hodge theory (as opposed to rational p-adic Hodge theory)? Answers which understand Hodge theory as the study of Galois stable $\mathbb{Z}_p$-...

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179 views

### Hochschild Homology and Formal Geometry

My question concerns the degeneration of the spectral sequence computing Hochschild homology of differential operators on a smooth affine variety $X$.
The spectral sequence arises from the ...

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**1**answer

144 views

### Multiple mirrors phenomenon from SYZ and HMS perspective

There is a set of ideas called mirror symmetry which, roughly speaking, relates symplectic and complex geometry of Calabi--Yau manifolds. There are also extensions to Fano and general type varieties ...

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91 views

### Berthelot’s comparison theorem and functoriality

Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$.
Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...

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172 views

### Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?

Please let me know whether this question is suitable for Mathoverflow.
Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. ...

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309 views

+50

### Are nearby points in an algebraic curve necessarily connected?

I would like a result of the following form:
For every algebraic curve $C$ in $\mathbb{R}\mathbf{P}^{n-1}$, there exists an
explicit and easy-to-compute $\epsilon=\epsilon(C)>0$ such that ...

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**1**answer

148 views

### rank of Jacobian of Fermat curve and Chabauty-Coleman method

Consider the fermat curve $F(p)$ over $\mathbb Q$ which is the projective closure of $X^p+Y^p=1$ inside projective plane, where $p$ is a prime number and without loss of generality we assume $p>2$. ...

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156 views

### p-adic Hodge theory for singular projective varieties

In p-adic Hodge theory, one has comparison theorems relating, for example, the crystalline cohomology of the special fiber of a smooth proper family with the etale cohomology of the rigid-analytic ...

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**1**answer

82 views

### Linear operator on polynomials and invariant sets of roots

Let $T:\mathbb{C}_n[x] \to \mathbb{C}_n[x]$
be a linear map from the vector space of polynomials of degree $n$ to itself.
Let $S \subset \mathbb{C}$ be a set with at least $3$ points, such that
for ...

**4**

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**1**answer

134 views

### Morphism of sites and abelian sheaf cohomology

Let $f : \mathcal{C}\to\mathcal{D}$ be a morphism of sites (see the Stacks Project) with induced morphism of topoi
$$(f^{-1}, f_*) : Sh(\mathcal{D})\to Sh(\mathcal{C}).$$
By assumption, $f^{-1}$ is ...

**13**

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**1**answer

481 views

### GAGA for henselian schemes

In this paper, F. Kato recollects basic facts on henselian schemes and proves some partial results towards GAGA in the context of henselian schemes.
Let $I$ be a finitely generated ideal in a ...

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**1**answer

84 views

### Core components of quiver varieties as fiber bundles of flag varieties

Is there an example of Nakajima quiver variety of type A which has all core components smooth, such that at least one of them is NOT an iterated fibre bundle of flag manifolds (i.e. a space obtained ...

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**1**answer

96 views

### Complete intersection subvariety of projective variety

I am not able to find any literature which studies complete intersection subvarieties of a projective variety, all good references consider CI in projective space.
My guess is, a subvariety X of ...

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**1**answer

263 views

### Where to find some subset of Khovanskii's 15 proofs of the BKK theorem?

(I asked this question on MSE, but someone suggested it would be better asked here.)
I'm a fan of the Bernstein-Khovanskii-Kushnirenko theorem (that the number of solutions in $(\mathbb{C}^*)^n$ to a ...

**9**

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**2**answers

639 views

### Homologicaly nice commutative rings

Hi,
Let $R$ be a commutative regular local ring. Is it true that for every $p \in Spec(R)$ there is a finitely generated $R$-module $M_p$ such that projdim($M_p$) = ht($p$) and Ass($M_p$) = {$p$}?
...

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82 views

### Factorizations of etale morphisms

Let $f:X \rightarrow Y$ be a finitely presented separated etale morphism, with $Y$ quasicompact and quasiseparated.
By Zariski’s main theorem, we can factor $f$ as $f= g \circ j$ with $j$ an open ...

**3**

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107 views

### Quotient of a smooth projective surface by an involution

Is the quotient of a smooth complex projective surface by an involution projective? Suppose the quotient happens to be smooth; does that change the situation?

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**1**answer

271 views

### A clarification of an argument in “Perfectoid spaces”

The page 50 of (the arXiv version of) the above-mentioned paper of P. Scholze says "Now the Poincare duality pairing implies that $H^i(Y_{\mathbb{C}_p, et}, \bar{\mathbb{Q}}_l)$ is a direct summand of ...

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109 views

### Sources of derived schemes in geometric representation theory

What are some derived schemes naturally arising in geometric representation theory?
Some examples include:
Steinberg scheme
Hilbert scheme
Moduli stack of local systems.
Now, this looks like a ...

**7**

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**2**answers

125 views

### The degree of the hypersurface of pfaffian cubic fourfolds

Let $\Pi:=\mathbb{P}(H^0(\mathbb{P}^5,\mathcal{O}_{\mathbb{P}}(3)))$ be the space of cubic fourfolds in $\mathbb{P}^5$. It is well-known that those cubics which are pfaffian, i.e. defined by the ...

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115 views

### Are Chain Complexes Related to the Tangent Bundle Construction?

For a scheme $X$ over $\text{Spec}(K)$, we can consider maps $\text{Sch}(\text{Spec}(K[d] / d^2), X)$, which we can think of as the tangent bundle over $X$. A map $\text{Spec}(K) \rightarrow S$ picks ...

**56**

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**16**answers

11k views

### Good introductory references on algebraic stacks?

Are there any good introductory texts on algebraic stacks?
I have found some readable half-finsished texts on the net, but the authors always seem to give up before they are finished. I have also ...

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112 views

### Construction of Fano threefold of degree $5$ and its defining equations

The Fano threefold $X$ of index $2$, degree $5$ and Picard number $1$ is known to be a general codimension $3$ linear section of the $Pl\ddot{u}cker$ embedding of Gr(2,5).
My first question: what ...

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130 views

### Schemes obtained replacing variables by $n \times n$ matrices?

Let $k$ be an algebraically closed field and $n \geq 1$ be an integer. Choose a polynomial $f \in k[x_1,\dots,x_m]$ and place the variables in a fixed order (i.e we choose a preimage of $f$ inside $k\...

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147 views

### Regular functions vs holomorphic functions

Let $X$ be an affine smooth variety over the complex numbers, $X^{an}$ its associated smooth complex analytic space, and $\mathcal{O}$, resp. $\mathcal{O}^{an}$ the respective structure sheaves.
Is ...

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133 views

### Grothendieck, Sphere, Cylinder [closed]

Archimedes' theorem relating the surface areas and volumes of the sphere and cylinder is well known. I'm wondering if scheme theory (specifically etale cohomology of schemes) may be useful to gain ...

**3**

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221 views

### Restriction of Ext sheaves on closed subschemes

Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve. For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of the fiber of $f$ over $t$, and let $\mathcal{F}$ a coherent ...

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68 views

### Minimal size of an open affine cover for an open complement

Let $X$ be a smooth projective scheme and $Y$ be a projective subscheme of $X$, not necessarily smooth. Are there any known results about the minimal size of an open affine cover (number of affines in ...

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**1**answer

242 views

### Definition of simple linear algebraic group

Why is it that many sources define simple (or almost-simple) linear algebraic group $G/k$ to be a connected, semisimple linear algebraic group such that every proper connected normal subgroup is ...

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82 views

### Linear projection from a point preserves flatness

Let $\pi:\mathcal{X} \to S$ be a flat family of affine curves contained in $\mathbb{C}^n$ for $n \ge 3$ i.e., $\mathcal{X} \hookrightarrow \mathbb{C}^n_S$ and the inclusion commutes with the natural ...

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212 views

### Deligne Pairing v.s. Weil Pairing on a Family of curves

We have the Deligne Pairing on a family of curve $\pi:X\to S$ by using
$$\langle L,M\rangle_{\mathrm{Pic}^0(X/S)}=\det R\pi_*(L\otimes M) \otimes (\det R\pi_*L)^{-1}\otimes (\det R\pi_*M)^{-1} \otimes ...

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276 views

### How does multiplication affect degrees?

Let $M(n) \sim \mathbb{A}^{n^2}$ be the space of $n$-by-$n$ matrices, seen as an affine space over a field $K$, and endowed with the usual matrix multiplication. Let $V$ and $W$ be subvarieties of $M(...

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506 views

### stackification commutes with finite limits?

Suppose we work on the Grothendieck site $\mathcal{C}$ of all schemes in the fpqc topology. If it helps it is also fine with me to work only over affine schemes.
Let us denote the category of stacks ...

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76 views

### Degeneration of cycle class map

Let $f:\mathcal{X} \to \Delta$ be a flat family of projective varieties, smooth over the punctured disc $\Delta^*$ and the central fiber is a simple normal crossings divisor. Let $\mathcal{Z} \subset \...

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**2**answers

847 views

### Weil's paper under a pseudonym on deforming singular varieties

I am looking for a paper of Weil that is published under a pseudonym, in which he proves a statement along the lines of: a singular algebraic variety cannot be deformed into a nonsingular one.
Thanks ...

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136 views

### An inverse problem for Grothendieck rings of varieties

Suppose $A$ is a given commutative ring, and suppose that one knows that $A$ is isomorphic to the Grothendieck ring of $k$-varieties for some unknown field $k$.
Can $k$ be recovered from $A$ ? If ...

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**1**answer

163 views

### Is the subgroup generated by a conjugacy class of semisimple elements Zariski closed?

Let $k$ be an arbitrary field with $\operatorname{char}(k) \neq 2$. Let $G$ be a linear algebraic group over $k$. Let $X$ be the conjugacy class of a semisimple element $s \in G(k)$ of order 2 (or a ...

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48 views

### Computing Gröbner basis elements of some constant degree

I'm wondering if there is any way or any special set of ideals such that there is an efficient way to compute elements of degree at most $d$ in a Gröbner basis for that ideal.
If you have any paper ...

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2k views

### Definition of relative Picard functor

Let $X \to S$ be a morphism of schemes. The relative Picard functor from schemes over $S$ to abelian groups is usually defined by the formula $T \mapsto \text{Pic}(X \times_S T)/p^{*}\text{Pic}(T)$, ...

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211 views

### What do non-principal divisors in a Picard group look like

The way Divisors on Elliptic Curves are motivated in cryptography is to say it's a convenient way to represent rational functions by keeping track of multiplicities of the zeros and poles of a ...