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

**5**

votes

**1**answer

66 views

### Regular minimal model of $X_0(p^2)$

Consider the compactified modular curve $X_0(p^2)$ and the corresponding algebraic curve over $\mathbb{Q}$. My questions are the following:
Where do the cusps of $X_0(p^2)_{\mathbb{Q}}$ live? That ...

**0**

votes

**0**answers

51 views

### Finding stable ideals of $\mathbb{F}_3[[X,S]]$ by group action

Let $k > 1$ be a positive integer and define the action $\sigma_k$ on $\mathbb{F}_3[[X,S]]$ by:
$\sigma_k: X \mapsto X + S + X^k$
$\sigma_k: S \mapsto S + S^3$.
Conjecture: There exists a ...

**0**

votes

**0**answers

108 views

### Existence of rational curve and semi-positivity of bisectional curvature

Let $X$ be a compact Kahler manifold with Kahler form $\omega$ such that there exists a rational curve $C\subset X$, then bisectional curvature of the Kahler metrics $\omega$ is semi-positive?. What ...

**4**

votes

**0**answers

67 views

### Is there an explicit construction of Tate-Beilinson residue?

Background: Tate's construction of abstract residues is generalized in Beilinson's constructon as follows:
Let $E$ be a unital, cubically decomposed $k$-algebra, i.e., (a) there are two sided ideals ...

**55**

votes

**1**answer

2k views

### Did Bourbaki write a text on algebraic geometry?

Certainly Bourbaki never wrote an introduction to algebraic geometry: we would have heard about it, right?

**2**

votes

**0**answers

144 views

### the origin of the differential of Spectral Sequence? [on hold]

I'm wondering why the differential in homological Spectral Sequences $(E^*_{p,q},d^r)$ is defined as;
$d^r_{p,q}: E^{r}_{p,q} \rightarrow E^{r}_{p-r,q+r-1}$ .
From Weibel's homological algebra(p122), ...

**7**

votes

**1**answer

460 views

### Analytic continuation for $L$-functions of elliptic curves

Let $E$ be an elliptic curve over a number field.
When $E$ has no CM and is a $\mathbb Q$-curve (i.e. it is $\overline{\mathbb Q}$-isogenous to all of its conjugates), it is nowadays known that $E$ ...

**0**

votes

**0**answers

175 views

### Quotient by a proper equivalence relation

Let $X$ be a scheme and $R$ be a proper equivalence relation on $X$.
What can be said about the geometric structure of the quotient $X/R$?
Is it representable by a stack, for example?

**1**

vote

**0**answers

233 views

### How to Construct a ''Nice'' Birational Model in Characteristic $p>0$?

Let $X=Spec\ A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ is a prime Weil-divisor on $X$.
Now, I ...

**6**

votes

**2**answers

167 views

### $G$-invariant holomorphic vs. polynomial functions

Let $X\subseteq\Bbb C^n$ be a smooth affine variety and $G$ a complex reductive group acting algebraically on $X$.
Let $x_0\in X$. If there is a non-constant $G$-invariant holomorphic function ...

**1**

vote

**1**answer

384 views

### Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.

I would like to know a reference of the following statement (or counter example).
Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is ...

**2**

votes

**0**answers

93 views

### Semistability of a sheaf on nodal curve

Suppose $X$ is a projective, connected, nodal curve (can be reducible) over an algebraically closed field $k$ of arbitrary characteristic. Let $F$ be a pure sheaf on $X$ and denote by $\pi^{*}(F)$ its ...

**0**

votes

**0**answers

154 views

### Can the zero-degree part of $M_f \otimes_{S_f} N_f$ be identified with $M_{(f)} \otimes_{S_{(f)}} N_{(f)}$?

The isomorphism ${(M \otimes _ {S} N)} _ {f} = M _ {f} \otimes _ {S _ {f}} N _ {f}$ is well-known. Here, $S$ is a graded ring, and $M,N$ are graded $S$ modules.
Now, let $f$ be any homogeneous ...

**0**

votes

**0**answers

141 views

### Is sum $(E_i, E_j)$ non-positive, with $E_i$'s the exceptional components of a desingularization

Let $Y$ be an integral normal 2-dimensional scheme and let $X\longrightarrow S$ be a flat projective morphism, where $S$ is a Dedekind scheme.
Let $f:X\longrightarrow Y$ be a minimal resolution of ...

**2**

votes

**0**answers

82 views

### Kunneth decomposition of the relative diagonal of a projective bundle

Let $\mathcal{E}$ be a projective bundle of rank $r$ over a smooth complex quasi projective variety $B$, and form its associated projective bundle $\chi :=\mathbb{P}(\mathcal{E})$. Let $\pi : \chi ...

**4**

votes

**3**answers

125 views

### Birationally transforming a quartic elliptic curve

Consider the elliptic curve
$$y^2=ax^4+cx^2+dx+f$$
I am aware that there are algorithmic methods for birationally transforming a nondegenerate cubic curve into the Weierstrass canonical form ...

**5**

votes

**1**answer

105 views

### Density of non-algebraic leaves in the characteristic foliation

Let $X$ be a compact complex manifold equipped with a holomorphic symplectic form $\omega$. Let $D$ be a smooth divisor on $X$. At each point of $D$, the restriction of $\omega$ to $D$ has ...

**5**

votes

**1**answer

157 views

### Bounding the degree of an algebraic extension containing solutions to polynomials

Also posted on math.stackexchange...
Let $F$ be a field, and let $f_{1},\ldots, f_{s}$ be polynomials in $F[x_{1},\ldots, x_{t}]$. Assume that the degree of the polynomials is bounded by $d$, by ...

**3**

votes

**1**answer

185 views

### Maximal length of filter regular sequence

Let $A$ be a Noetherian ring and $R$ is a standard graded ring over $A.$ Let $M$ be a finitely generated graded $R$-module and $I$ be a graded ideal of $R.$ Then $x_1,\ldots,x_r\in I$ is called ...

**0**

votes

**0**answers

113 views

### local analyticity of volumes of slices of semi-algebraic sets

I would like a reference and/or a simple proof using well-known results of the following, which I think is true. (If it's false, I'd like to know that as well of course -- and ideally a way to modify ...

**1**

vote

**1**answer

161 views

### harmonic analysis on $p$-adic $x^2 + y^2 + z^2 = 1$?

Are there p-adic analogues to spherical harmonics? In the case of $K = \mathbb{R}$, the spherical harmonics form a basis to $L^2 [SO(3)]$ where
What happens in the $p$-adic case? Is there sphere ...

**2**

votes

**0**answers

117 views

### Are these moduli problems of curves “well-behaved”?

Let X be a smooth projective surface over $\mathbb C$, and let $d\geq 3$ be an integer. Suppose that all smooth hypersurfaces of degree $d$ are of genus $g\geq 2$.
Let $H_{X,d}$ be the Hilbert scheme ...

**17**

votes

**0**answers

453 views

### Smooth curves on smooth varieties

Let $X$ be a smooth, proper algebraic variety over a field $k$, of positive dimension.
Is it true that $X$ contains a smooth Zariski-closed curve?
If it is projective, this is true by Bertini. But ...

**0**

votes

**0**answers

181 views

### Existence of flat models of a smooth finite type algebra over $R((t))$

Let $k$ be a field, $R$ a $k$-algebra (of finite type if necessary),
$B$ an algebra of finite type over ring of the formal Laurent series $R((t))$, which is smooth.
Up to this generality, can one ...

**2**

votes

**1**answer

131 views

### projectivity with assumption of big and semi-amplness

Let $X$ be a compact Kaehler manifold with $D$ be an effective divisor on $X$ such that $K_X+D$ is semi-ample and big then $X$ is projective?

**1**

vote

**0**answers

233 views

### Existence of Rational curve

Let $X$ be a compact Kähler manifold such that $K_X$ is nef. When there is no rational curve on $X$?

**2**

votes

**1**answer

115 views

### No cuspidal character sheaves on GL(n)

We need a reference for the fact that there are no cuspidal character sheaves on $GL_n$ unless $n=1$.
See page 11 of http://www.kurims.kyoto-u.ac.jp/~arakawa/Henderson_mgsctalk2.pdf.

**4**

votes

**1**answer

300 views

### Normalization of complete intersection

Let $A$ be an integral complete local ring over a field which is complete intersection.
Let $B$ be a normalization of $A$.
Q. Is $B$ Gorenstein?
I guess that even the normalization of Gorenstein ...

**0**

votes

**1**answer

368 views

### Can monodromy be described by the same matrix for chosen generators in case of the same singularity type?

Let $X$ be a surface in $\mathbb{P}^3$. We have a fibration $f: X \longrightarrow \mathbb{P}^1$, and $f^{-1}(s_1)$ and $f^{-1}(s_2)$ have the same singularity type. Let $\gamma_1$ and $\gamma_2$ be ...

**10**

votes

**3**answers

1k views

### Zariski tangent spaces to representation varieties

In Bill Goldman's paper "The Symplectic Nature of the Fundamental Groups of Surfaces" (Advances, 54, 200-225, '84) it is stated that the "Zariski tangent space" to a representation space Hom$(\pi, ...

**2**

votes

**2**answers

246 views

### Epimorphisms between affine group schemes

Does there exist a simple characterization of epimorphisms between affine group schemes over a field ? Are they faithfully flat morphisms ?

**7**

votes

**0**answers

120 views

### $\Bbb A^1$-Localisation of Schemes, and $\Bbb A^1$-Rigid Schemes

Question 1: Are there some publications or preprints that provide $\Bbb A^1$-fibrant replacements of certain classes of (smooth) schemes?
Of course, smooth schemes that are $\Bbb A^1$-fibrant are ...

**4**

votes

**1**answer

560 views

### What sort of ind-scheme is this?

It apparently follows from work of Velu (MathSciNet) that every isogeny between elliptic curves in (long) Weierstrass form over $k$ can be written in the form
$$
\left(\frac{u(x)}{v(x)}, ...

**9**

votes

**1**answer

659 views

### Is there an algebraic description of Yang-Mills measure?

Consider $X=\operatorname{Hom}(\pi,\mathrm{SL}(2,\mathbb C))/\! \!/\mathrm{SL}(2,\mathbb C)$ and $Y=\operatorname{Hom}(\pi,\mathrm{SU}(2))/\mathrm{SU}(2)$, where $\pi$ is a surface group. Note that ...

**-1**

votes

**1**answer

133 views

### Invariance of the fiber-dimension of a finite map

Let $A\subseteq B$ be commutative Noetherian rings such that $A$ is a regular ring, i.e., $A_{\mathfrak{m}}$ is a regular local ring for all maximal ideals $\mathfrak{m}$ of $A$ and $B$ is a finite ...

**0**

votes

**0**answers

70 views

### Moebius linear fractional transformation in higher dimensions [closed]

A linear fractional transformation is a function of the form $f(z)=\frac{az+b}{cz+d}$ where $a,b,c,d \in\mathbb{C}$. If $ad-bc\neq0$ then $f$ is called Möbius transformation. The question is if there ...

**5**

votes

**1**answer

366 views

### Holomorphic vector bundles over $\mathbb{C}^{n}\setminus 0$

Is it true that every holomorphic vector bundle over $\mathbb{C}^{n}\setminus 0$ is trivial? If not true, how can one construct a counterexample?
And just a small note here (wrong):
For $n\leq 2$, ...

**15**

votes

**1**answer

1k views

### What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?

**1**

vote

**1**answer

181 views

### Algebraicness of trace field of finite volume hyperbolic 3-manifold and dimension of $\mathrm{SL}(2,\mathbb{C})$-character variety

Does the following statement:
"Let $G$ be a finitely generated
group and let $X(G)$ be the
$SL(2,\mathbb{C})$-character variety
of $G$. Suppose $X(G)$ contains an
irreducible component ...

**4**

votes

**1**answer

224 views

### What are Motivic homotopy types?

There are suggestions that says that Grothendieck developed (in some sense) a theory of Motivic homotopy types or at least named it.
I would like to know the reference in which Grothendieck did it, ...

**3**

votes

**1**answer

162 views

### Extending a prime divisor to a principal divisor

Let $X$ be a noetherian,integral,separated scheme which is nonsingular in codimension $1$. Let $Z_1, \ldots, Z_k$ be a fixed set of prime divisors. Now given a prime divisor $Y$, does there exist ...

**11**

votes

**1**answer

409 views

### Elements of arbitrary large order in the first Galois cohomology of an elliptic curve

Let $E$ be an elliptic curve over $k=\mathbb{Q}$. Consider $H^1(k,E)$.
In this answer Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I ...

**3**

votes

**1**answer

210 views

### When is an irreducible $\mathrm{SL}_2(\mathbb{C})$ representation of a cusped hyperbolic 3-manifold scheme reduced or smooth?

Let $M$ be an orientable cusped hyperbolic 3-manifold. Let $\rho \in \mathrm{Hom}(\pi_1(M),\mathrm{SL}_2(\mathbb{C}))$ be an irreducible representation.
Is $\rho$ scheme reduced ?
What can ...

**7**

votes

**4**answers

809 views

### Clarification on the weak BSD conjecture

It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to ...

**4**

votes

**0**answers

205 views

### Construction of Dualizing sheaf

I was going through the construction of dualizing sheaf given in Hartshorne [III, 7, Lemma 7.4]. The proof apparently omits lots of details. In particular it does not mention any of the $i^* , i_*, ...

**3**

votes

**0**answers

55 views

### Psi-classes on moduli spaces of weighted curves

Let $\overline{\mathcal{M}}_{g,A[n]}$ be the stack of weighted genus $g$ curves with weights $A[n]=(a_1,...,a_n)$, and let $\pi:\mathcal{C}\rightarrow \overline{\mathcal{M}}_{g,A[n]}$ be the universal ...

**4**

votes

**2**answers

322 views

### An apparent equivalence of the category of affine schemes over $S$ and the category of quasi-coherent $\mathcal{O}_S$-algebras

I had asked something very similar before on math.se (deleted now) but unfortunately it hadn't received a lot of attention. I decided to re-ask here.
Let $S$ be a fixed scheme. Is the following true?
...

**4**

votes

**1**answer

121 views

### Minimal Nagata-like compactification

Working over a field $k$, Nagata's compactification theorem implies that any separated scheme $X$ of finite type over $k$ admits a compactification (a dense open immersion $i \colon ...

**1**

vote

**1**answer

241 views

### Irreducible variety

I asked a similar question at MSE, as the question seemed quite basic to me, but did not get any hint in 24 hours, except for one upvote for the question itself. I still think I am stuck with some ...

**3**

votes

**1**answer

230 views

### How to tell if it's a Moishezon morphism

Suppose that $f \colon X\rightarrow S$ is a proper morphism of reduced and irreducible complex spaces and $f$ is a smooth deformation in the sense of Kodaira and Spencer. If we know each fiber $X_s$, ...