# Tagged Questions

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

**-3**

votes

**0**answers

25 views

### If a family of complex analytic space is smooth

If I have a deformation in complex analytic space setting. The parametrize space is U, total space is X. If U is smooth and every fiber X(t) is smooth, if X is smooth and the map from X to U is ...

**0**

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**0**answers

56 views

### Weierstrass model for Calabi-Yau fourfolds

The local Weierstrass model for a Calabi-Yau threefold looks like:
$$ y^2 = x^3 + f\left(z_i\right)x + g\left(z_i\right) \subset \mathbb{P}^2_{\left[x,y\right]} $$
With $f\left(z_i\right) \in ...

**0**

votes

**0**answers

38 views

### Zariski open set of linear forms

Let $I$ be a graded homogeneous ideal over $k[x_1, ... ,x_n]$ and $h$ a linear form, let $H$ be the corresponding hyperplane and $I_H$ the restriction of $I$ to $H$.
I am looking for a Zariski open ...

**1**

vote

**0**answers

48 views

### Open covering of the Hilbert functor of points

Let $R \to A$ be a homomorphism of commutative rings. Define the functor $$\mathrm{Hilb}^n_{A/R} : \mathsf{CAlg}(R) \to \mathsf{Set}$$ as follows: If $B$ is a commutative $R$-algebra, then ...

**0**

votes

**0**answers

55 views

### Albanese map on a universal jacobian associated to a curve on an abelian surface

Given an abelian surface $A$ and a curve $C$ on it, consider the component $\{C\}$ of the Hilbert scheme of curves with the same cohomology class of $C$.
I would say that it has a fibration structure ...

**0**

votes

**0**answers

99 views

### Cohomology group vs sheaf of cohomology group

Suppose $F$ is a coherent sheaf on a smooth (algebraic or complex) variety $X$. Then we can consider the cohomology groups $$H^p(X,F)$$ for all $i$. Now, let we consider the sheaf ...

**1**

vote

**0**answers

64 views

### Singularities of secant varieties of rational normal curves

Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper:
...

**2**

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**0**answers

54 views

### Which valuations of a field yield codimension $1$ subschemes of their 'models'

For a field $F$ (for example, a one generated by a finite number of its elements) there is a directed set of its 'models' (in this case those are 'arithmetic' schemes whose fraction field is $F$). It ...

**4**

votes

**2**answers

161 views

### When are isotrivial families split by a finite base-change?

A well-known theorem of Grauert and Fischer states that a smooth proper family of complex manifolds is a locally trivial fibration as soon as all the fibers are isomorphic. It is also easy to obtain ...

**1**

vote

**0**answers

64 views

### Question about the specialization map for Etale Fundamental Groups

Let $A$ be a complete, discrete valuation ring, and let $s$ (resp $\eta$) be the special (resp. generic) point of $S=Spec(A)$. Let $\phi:X \rightarrow S$ be a proper morphism and fix geometric base ...

**2**

votes

**1**answer

482 views

### is cross product of 2 or 3 vectors in 4 D is possible ? why ? i'm an engineer and please kelp me in explaining in a way i can understand? [on hold]

Is cross product of 2 or 3 vectors in 4 D is possible ? why ? i'm an engineer and please kelp me in explaining in a way i can understand ? It is also not clear that why vectors depend on the dimension ...

**1**

vote

**0**answers

61 views

### Obstruction to Gorenstein Liaisons of space curves

Let $P_1,P$ be Hilbert polynomials of curves in $\mathbb{P}^3$. Denote by $H_{P_1,P}$ the flag Hilbert scheme parametrizing pairs $(C_1 \subset C)$ where $C_1, C$ are of Hilbert polynomials $P_1$ and ...

**1**

vote

**0**answers

231 views

### Grothendieck's letter to serre

Is it the letter dated in 27/08/1965 of Grothendieck where he presents to Serre the Standard conjectures on algebraic cycles?

**1**

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**0**answers

68 views

### A topological criterion for connectedness of a semi-ample divisor

I have a half page long proof of the following statement, and I would like to know if this is (a corollary of) a well known statement. Maybe there is a reference or a three lines proof?
Statement. ...

**0**

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**0**answers

39 views

### Embedding dimension: local finiteness & intuition for more general spaces

Can every complex analytic space be covered by Stein spaces of finite embedding dimension?
I am almost sure that ought to be true. Here the definition of embedding dimension I have in mind is
$$
...

**1**

vote

**1**answer

130 views

### Examples of nontrivial local systems in Decomposition Theorem

There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that
$$Rf∗IC_X \cong \oplus_a ...

**2**

votes

**3**answers

431 views

### Reference request: SGA7

I want to read SGA7. Without considering the others SGA and EGA, Which are the textbooks for monodromy theory?

**8**

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**0**answers

113 views

+50

### Newton polygons of modular polynomials

This is pretty much straightforward curiosity. Is there anything known about Newton polygons of classical modular polynomials (polynomial relations between $j(\tau)$ and $j(n\tau)$)? I understand that ...

**5**

votes

**1**answer

184 views

### Is the Tate conjecture known for etale covers of products of curves

Let $X$ be a (smooth projective geometrically connected) surface over a finitely generated field $k$. The Tate conjecture predicts that, for $l$ a prime number invertible in $k$, the Chern class map ...

**12**

votes

**2**answers

457 views

### History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...

**9**

votes

**1**answer

169 views

### Borel-Serre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$

Let $\mathbb{H}^3$ be the three-dimensional hyperbolic space. Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Then $SL_2(\mathcal{O}_K)$ acts on $\mathbb{H}^3$ ...

**2**

votes

**0**answers

124 views

### Universal cover of elliptic curve without a point

This is something that I perhaps would be expected to know, but don't. Let $E_\tau$ be the elliptic curve ${\mathbb C}/({\mathbb Z +\mathbb Z} \tau)$. Consider the complement of a point in $E_\tau$,
...

**2**

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**0**answers

65 views

### Degree of join of two varieties

Let $X\subset\mathbb{P}^n$ be an irreducible variety of degree $d$ and dimension $n-4$. Let $L\subset X$ be a line of mulitplicity $m$ in $X$. Assume that $m+1\leq d$ so that the general line spanned ...

**2**

votes

**2**answers

167 views

### Example of linearization for GIT

Take a vector space $V$ (finite dimensional, over the complex numbers), let $G=SL(V)$. The group $G$ acts on $\mathbb{P}V$ and we can linearize its action to an action on the line bundle ...

**4**

votes

**1**answer

223 views

### Non-vanishing of elements in cohomology of full Flag varieties

Consider the full flag variety $F_n$ consisting of full flags in $\mathbb C^n$. There is a collection of tautological bundles on $F_n$:
$0=U_0\subset U_1\subset ...\subset U_{n-1}\subset U_n=\mathbb ...

**2**

votes

**2**answers

207 views

### Basic questions on the Hilbert scheme/ Douady space

Let $X$ be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of $X$. More ...

**9**

votes

**3**answers

1k views

### Weil's Riemann Hypothesis for dummies?

Weil's Riemann Hypothesis is a deep result that I don't fully understand, but it has understandable corollaries which interest me. For example:
(a) For any projective curve $X$ satisfying certain ...

**2**

votes

**2**answers

262 views

### Canonical Sheaf of Projective Space

I am stuck on one step that occurs without explanation in several Algebraic geometry books.
Starting from the exact sequence
$$0\rightarrow \Omega_{\mathbb{P}^n}\rightarrow ...

**1**

vote

**2**answers

199 views

### commutative algebra, diagonal morphism

can anyone help me with the following statement (it is part of a bigger proof where it is not explained).
Let $B$ be a finite type $A$-algebra and consider the kernel $I$ of the diagonal ...

**2**

votes

**0**answers

50 views

### calculating the Hilbert polynomial of a scheme given its primary decomposition

Given a scheme $X$ in $\mathbb{P}^n$, let $I_X$ be its, saturated, associated ideal. Suppose that a primary decomposition of this ideal is given by
$$
I_X =I_1 \cap \ldots \cap I_2
$$
I was ...

**0**

votes

**0**answers

411 views

### A letter from J. P. Serre

Which is the letter where J. P. Serre present "Analogues Kählériens de certaines conjectures de Weil" to Weil?

**4**

votes

**2**answers

268 views

### Time-line until the publicaton of Weil of “Numbers of solutions of equations in finite fields”

In "On the history of the Weil Conjectures" Dieudonné says:
"Appropriately enough, the story, as with so many problems in number theory, begins with Gauss...".
C. F. Gauss, Disquisitiones ...

**1**

vote

**0**answers

277 views

### Rational structures on the flag variety over a finite field

Some Notions
A variety over a field is defined to be a scheme of finite type over this field.
An $\mathbb{F}_q$-rational structure of an $\bar{\mathbb{F}}_q$-variety $V$ is a $\mathbb{F}_q$-variety ...

**1**

vote

**1**answer

101 views

### Perfectness of the Jacobian of a curve

Let $C$ be a smooth projective curve over a field $K$ of characteristic $0$ (but not necessarily algebraically closed). Let $\mathcal{L}$ be a line bundle on $C$ of degree $0$. Fix an integer ...

**0**

votes

**3**answers

211 views

### question about the induced homomorphism of etale fundamental groups

Background/Setup
For any connected scheme $S$, let $\text{FEt}_S$ denote the category of finite etale $S$-schemes. Let $f : X\rightarrow Y$ be a morphism of connected schemes, then for any finite ...

**2**

votes

**0**answers

170 views

### p-adic etale cohomology

Let $X$ be a smooth projective scheme over $\mathbb{Z}_p$, with special fiber $X_s$ over $\mathbb{F}_p$, generic fiber $X_{\eta}$ over $\mathbb{Q}_p$, and geometric generic fiber $\bar{X_{\eta}}$ over ...

**7**

votes

**0**answers

443 views

### Homotopy type of complex algebraic varieties

In his 1974 ICM adress "Poids dans la cohomologie des variétés algébriques", Pierre Deligne explains that any finite polyhedron has the same homotopy type as a complex algebraic variety (section 6.).
...

**3**

votes

**1**answer

199 views

### Second betti number of compact analytic spaces

Let $V$ be a proper singular complex algebraic variety, possibly nonprojective ($dim(V)=n>0$). I would like to know:
1) if its second Betti number is non zero,
2) same question but now $V$ is a ...

**2**

votes

**0**answers

120 views

### What is the relation between Beilinson's conjectures and Standard conjectures of algebraic cycles?

Do Standard conjectures on the K-theory of varieties over finite field have implications in the motivic cohomology of Z where exist the correct formalism of Beilinson's conjectures?
What is the ...

**2**

votes

**1**answer

162 views

### Castelnuovo Mumford Regularity

Let
$$
0 \rightarrow E_{n-1} \rightarrow ... \rightarrow E_1 \rightarrow E_0 \rightarrow I \rightarrow 0 ,
$$
be a minimal free resolution of ideal
$I$,
where
$$
E_p = ...

**3**

votes

**0**answers

105 views

### Are there any useful Grothendieck topologies for which the H1 of $GL_n$ is not the set of rank $n$ vector bundles

Let n be a positive integer and X a scheme. Then for all the Grothendieck topologies I know (Zariski, etale, fppf) the set $H^1(X,GL_n)$ is the set of (isomorphism classes of) rank $n$ vector bundles.
...

**5**

votes

**0**answers

367 views

### Are there workable algebraic geometry approaches for the pentagon equation?

A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring.
A fusion ring is given by a finite set of integer ...

**2**

votes

**0**answers

84 views

### How does one show that slope stability of a vector bundle is an open condition with respect to the polarisation?

I would like a source for the following result, which I expect to be true (probably well known):
Let $X$ be a complex projective variety, $L$ an ample line bundle and $E$ a slope stable vector bundle ...

**9**

votes

**2**answers

614 views

### GIT over integers

Let $G$ be a reductive algebraic group over ${\mathbb Z}$ (or a finite
localization of a ring of integers $R$ in a number field) acting on an
affine scheme of finite type $M=Spec(A)$ over $R$.
We ...

**1**

vote

**2**answers

154 views

### Small birational maps and singularities of the pair

Let $f:X\dashrightarrow Y$ be a small birational map, where $X,Y$ are normal $\mathbb{Q}$-factorial varieties. Let $\Delta_X\subset X$ be an effective $\mathbb{Q}$-divisor such that the pair ...

**1**

vote

**0**answers

115 views

### Can Kuranishi families glue together to give a moduli space?

In his article "Moduli spaces of vector bundles on K3 surfaces and symplectic manifolds", Mukai constructs moduli of simple sheaves on a K3 surface through a theorem that asserts the existence of a ...

**2**

votes

**0**answers

82 views

### generalization Abhyankar's lemma

This question is related to a question I already asked on MO (smooth quotient out of a singular variety?), but I realized later that the hypotheses where not precise enough in my former question.
Let ...

**4**

votes

**1**answer

193 views

### Continued fraction expansion of an algebraic number and its conjugates

Let $w$ be an element of a Galois extension $L:\mathbb{Q}$ such that $\text{Gal}(L/\mathbb{Q})=\langle g\rangle$ is cyclic of order $n$ (here $\mathbb{Q}$ is rationals). Suppose we know the continued ...

**1**

vote

**1**answer

90 views

### Boundedness of the number of curves negative on a varying big divisor

For a divisor $D$ on a smooth complex projective surface $X$, the stable fixed part is the maximal effective divisor $E$ which, for every $n \in \mathbb{N}$, is contained in every memeber of the ...

**5**

votes

**0**answers

113 views

### Is every Noetherian *connected* ring a quotient of a Noetherian domain?

This question is a strengthening of this question (answered negatively), and arose due to David Speyer's answer here.
Geometrically, this asks if every Noetherian connected affine scheme can be ...