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

**3**

votes

**0**answers

135 views

### Is the Veronese variety “enough” to describe all the $SL(V)$-orbits in $\mathbb{P}(\textrm{Sym}^dV)$?

I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts.
Let $V$ be a complex vector space of dimension $n$, ...

**5**

votes

**1**answer

145 views

### curve over higher dimensional basis with 0-dimensional locus of bad reduction

Is there an example of a flat proper relative curve $X/S$ with geometrically connected fibres and with $\mathrm{dim} S > 1$ and $S$ regular and connected with $0$-dimensional locus of bad reduction ...

**2**

votes

**0**answers

93 views

### Resolution of a birational map to get an isomorphism

Suppose that $f:S\rightarrow V$ is a regular map, where S is a compact complex surfaces and $V$ is an algebraic surface such that $f$ induces an isomorphism on the rational function field.
Under ...

**11**

votes

**2**answers

316 views

### Quotient rule, differential operator on a localization is well-defined, underlying geometry?

Using the quotient rule, we obtain that the notion of differential operator on a localization is well-defined:$$\mathcal{D}_A(B_f) \cong \mathcal{D}_A(B)_f.$$Here, $B$ is a commutative $A$-algebra, ...

**3**

votes

**1**answer

285 views

### If $F(x,y)$ is a polynomial which is not a square, then how often is the specialization $F(x,a)$ a square?

Suppose $F(x,y)$ is a polynomial in two variables over a field $K$, and $F(x,y)$ is not a square. When is $F(x,a)$ a square for $a\in K$? I would guess that Hilbert's Irreducibility Theorem might help ...

**3**

votes

**1**answer

244 views

### How do I find a smooth curve in $J(C)$ through the 2-torsion points?

Let $X$ be the Jacobian of a genus 2 curve over $\mathbb{C}$. Let $L=\mathcal{O}(nC)$, where n is an even number. Is it possible to find a smooth curve from $|L|$ which is fixed by the involution ...

**10**

votes

**1**answer

574 views

### Relation between Weil Conjecture and Langlands Program

Recently I read Gelbart's An Elementary Introduction To The Langlands Program, which explained the origin of the program, and this question came to me. For an elliptic curve over finite field, the ...

**3**

votes

**1**answer

182 views

### Let $X$ be a projective variety and let $Pic(X) \cong \mathbb Z$, then $(X, −K_X) $ is $K$-semistable

Is there any approach for the following conjecture?
Let $X$ be a projective Fano manifold and let $Pic(X) \cong \mathbb Z$, then $(X, −K_X) $ is $K$-semistable.

**4**

votes

**0**answers

125 views

### Is there an example of a holomorphic vector bundle whose Atiyah class vanishes and does not admit a flat connection?

Let $E\to X$ be a holomorphic vector bundle over a Kähler manifold. The vanishing of the Atiyah class, $At(E)=0$, is equivalent to the existence of a holomorphic connection on $E$.
Moreover, it is ...

**2**

votes

**2**answers

261 views

### H. Cartan's “Variétés analytiques complexes et cohomologie”?

Does anyone know where I might find an online version (for free or purchase, translated or in french) of this paper by Henri Cartan from 1953? I know it was published in Colloque sur les fonctions de ...

**3**

votes

**0**answers

160 views

### Correspondence between Serre and Tate on resolution of singularities

I am told that there is a 1961 correspondence between J-P. Serre and J. Tate on resolution of singularities in characteristic 0, where Serre discusses his attempts to disprove the latter using R. ...

**2**

votes

**1**answer

89 views

### Sections of a sheaf of differentials on a weighted complete intersection

Let $X\subset\mathbb{P}(a_0,...,a_N)$ be a smooth $n$-dimensional weighted complete intersection in a weighted projective space $\mathbb{P}(a_0,...,a_N)$.
Is it true that if $q\geq 1$ then ...

**2**

votes

**0**answers

54 views

### $\theta$-characteristics and Heisenberg action

Let $X$ be a complete smooth curve over $\mathbb C$ and $A=Jac[2]$ be the group of 2-torsion elements in the Jacobian of $X$. Is their a relation between:
1) The $A$-torsor of ...

**3**

votes

**1**answer

195 views

### To what extent are toric manifolds and principal torus bundles “the same thing”?

I am a little confused by the different definitions for toric manifolds/varieties. Depending on the definition of toric manifolds and principal torus bundles that one chooses, when is a toric manifold ...

**5**

votes

**2**answers

196 views

### global section of affine $C^\infty$-scheme

I'm reading Algebraic Geometry over $C^\infty$-rings.
It is written that "If $\mathfrak{C}$ is not finitely generated then $\Phi_{\mathfrak{C}}:\mathfrak{C}\rightarrow ...

**0**

votes

**0**answers

39 views

### A quick question from the paper “Generalizations of reductions and mixed multiplicities” by D. Rees

In the proof of Theorem 1.3 in the paper "Generalizations of reductions and mixed multiplicities" by Rees here, Is it necessary to consider the ring $Q'=Q/{\bigcup\limits_{q \geq 1}(0:{(I_1\dots ...

**18**

votes

**2**answers

698 views

### how do you visualize characteristic class?

For cohomology, there are some equivalent definitions when the object we consider is sufficiently nice. Since I mainly work with algebraic variety, I will restrict the objects I am considering to be ...

**4**

votes

**0**answers

78 views

### Minimal discriminant of an elliptic curve in terms of its Galois representation

From the Galois representation of an elliptic curve $E$ we can read the conductor of $E$, and further some information about the minimal discriminant. So is there any more information about the ...

**3**

votes

**0**answers

84 views

### What happens to the gonality under a finite morphism of curves

Let $f:C\longrightarrow C'$ be a finite degree 2 morphism of smooth projective curves. If the gonality$(C')=k$, then we can say that the gonality$(C)\leq 2k$. Under what conditions is the gonality ...

**4**

votes

**2**answers

200 views

### Explicit $2$-descent on elliptic curves

Let $k$ be a field of characteristic $0$ and let
$$E: y^2 = f(x)$$
be an elliptic curve over $k$, with $\mathrm{deg}(f) = 3$. Kummer theory yields a map
$$\varphi:\mathrm{H}^1(k, E[2]) \to ...

**2**

votes

**0**answers

90 views

### Definition of order of vanishing of a polynomial

Let $X$ be a projective integral subscheme over an algebraically closed field $k$, regular in codimension 1. Then the different notions of order of the vanishing of $f$ at a codimension $1$ subscheme ...

**4**

votes

**0**answers

75 views

### Weil structures and rational structures on $\overline{\mathbb{Q}}_{\ell}$-sheaves

In the literature concerning characteristic functions of varieties over a finite field, there is a notion called Weil structure defined in the following way:
Definition Let $X$ be a finite type ...

**6**

votes

**0**answers

259 views

### Is Max (R) a Hausdorff space?

I asked the following question at < http://math.stackexchange.com/questions/1508168/is-max-r-a-hausdorff-space >, but I pose it here for any help.
Recall a space is totally disconnected if the ...

**5**

votes

**0**answers

165 views

### Semistable reduction and log structures

I have been reading Hyodo-Kato's paper on log-crystalline cohomology, and there is one statement there that has been troubling me.
To explain this, suppose we have a perfect field $k$ of ...

**2**

votes

**0**answers

64 views

### the normalized blowup

Let $X$ be a normal variety over $\mathbb{C}$ and $x\in X$ a singular point.
Let $f:Y^{\nu}\to X$ be the normalized blowup at $x\in X$. (i.e. $f$ is a composition of the blowup $Y:=Bl_xX\to X$ and ...

**8**

votes

**2**answers

300 views

### Are there varieties with non finitely generated Picard group and vanishing irregularity?

Let $X$ be a smooth projective variety over an algebraically closed field $k$.
Can it happen that $q(X) := \dim H^1(X,\mathcal O_X) =0$ and $\textrm{Pic} \,X$ is not finitely generated?
Certainly, ...

**0**

votes

**1**answer

78 views

### When can one infer degrees of generators of a ring from its hilbert series

I know that for a noetherian ring, it's hilbert series can be written as $$HS(t)=\frac{P(t)}{\prod_{i=1}^d{(1-t^{d_i})}}$$ where $P(t)$ is polynomial, and there are $d$ generators of degrees ...

**6**

votes

**3**answers

286 views

### Argument of Zariski density to prove rationality of a regular map

Question: I want to know if the following result is correct:
Let $k$ be a number field and $k_v$ be a completion of $k$ at some place $v$, denote $K_v$ an algebraic closure of $k_v$.
...

**17**

votes

**1**answer

476 views

### Reference for de Rham cohomology in positive characteristic

It is known in characteristic $0$ that (algebraic) de Rham cohomology is a Weil cohomology theory. However, in characteristic $p > 0$ it isn't, if only because it has mod $p$ coefficients, whereas ...

**2**

votes

**2**answers

160 views

### Quadratic twist of an elliptic curve given by Weierstrass model [closed]

What is the equation of quadratic twist of Weierstrass curve over prime field GF(p) if p mode 4 = 1 or 3 if we want to have isomorphism over GF(p)?
What is the equation for binary field ?
Thanks.

**4**

votes

**0**answers

131 views

### Proper base change for non-quasicoherent sheaves

For a proper flat map $f: X \to Y$ (of reasonable schemes) and a closed embedding $i: Y' \to Y$, we know by EGA the base change quasi-isomorphism, where $f'$ and $i'$ are the pullbacks of $f$ and $i$:
...

**3**

votes

**0**answers

82 views

### How to describe morphisms to a weighted projective space (bundle)?

The case of an usual projective space (bundle) is well known (Grothendiek,EGA II, Publ.Math. IHES, 8, 1961; or Hartshorne, Alg.Geom.).The more general case of toric varieties has been considered by D. ...

**4**

votes

**1**answer

133 views

### Semistability of principal bundle vs vector bundle

Ramanathan has defined the semistability of a principal $G-$bundle $E$ over a curve $X$ as follows:
$E$ is semistable iff for any parabolic subgroup $P\subset G$, for any reduction of the ...

**2**

votes

**1**answer

119 views

### Relation between curves in a complete linear system contained in another

Let $X$ be a projective surface over $\mathbb{C}$, let $x\in X$ be the only singular point of $X$. Let $L$ be an ample line bundle on $X$. Consider the blow up $Y$ of $X$ along $x$, ...

**10**

votes

**1**answer

281 views

### T-equivariant cohomology of flag variety

Let $X=G/B$ , where $G=GL_n(\mathbb{C}^n)$ and $B$ be the upper triangular matrices. I am curoius about the structure of $H^*_T(G/B)$ which I consider as a $H_T^*(pt)$-module. If we just consider ...

**2**

votes

**0**answers

103 views

### If G-invariant metric is always Kahler-Einstein

Suppose there is an Hermitian symmetric space of compact type X. It is realized in the following way:
$X\hookrightarrow\mathbb{P}^N$ and equip it with induced Fubini-Study metric g. What's more, the ...

**7**

votes

**0**answers

341 views

### Algebraic geometry introduction for homotopy theorists/algebraic topologists

Algebraic geometry has a plenty of decent introductory texts now. Some are of the classical commutative algebraic approach(following EGA), like Ravi Vakil's "Foundations".
Some use facts from ...

**14**

votes

**2**answers

535 views

### Algebraic surface of constant width?

Does there exist an irreducible polynomial $f \in \mathbb{R}[x, y, z]$ such that:
$$ V := \{ (x, y, z) \in \mathbb{R}^3 : f(x, y, z) \leq 0 \} $$
is a solid of constant width with a finite symmetry ...

**3**

votes

**2**answers

112 views

### Quartic symmetroids and 10-points sets

A quartic surface in $\mathbb{P}^3$ is said to be a "symmetroid" if its equation is obtained as the determinant of a 4x4 symmetric matrix of linear forms. It is well known that the general symmetroid ...

**1**

vote

**0**answers

73 views

### reference for weighted blow-up

Let $(0\in X)$ be a germ of a normal 3-fold with a singular point $0$
(over $\mathbb{C}$).
We think of $X$ as a small neighborhood of $0$ (for studying singularity).
If we can think $X$ as a ...

**2**

votes

**1**answer

139 views

### A curve in an abelian surface and its image in the Kummer surface

This is related to a question that I already asked Curve through the 16 singular points of a Kummer surface. I am new to Abelian varieties and I am trying to understand more about them.
Let $X=J(C)$ ...

**3**

votes

**1**answer

160 views

### Chevalley devissage

Let $G$ be an algebraic group over a perfect field $k$. Then it is know that it can be written as an extension of an affine algebraic group and a proper algebraic group.
Is there a similar result for ...

**3**

votes

**2**answers

216 views

### Curve through the 16 singular points of a Kummer surface

Let $X$ be an abelian surface over $\mathbb{C}$. Consider the Kummer surface $K$ associated to $X$, that is the quotient of $X$ by the action of involution on $X$, $x\mapsto -x$. Kummer surface is a ...

**4**

votes

**1**answer

155 views

### Neron-Severi group for product of curves

Let $C$ be a general genus $g$ curve, how can we describe the Neron-Severi group of its $n$-th self product $C^n=C\times \dots \times C$?
It is a lattice in $H^2(C^n,\mathbb{Z})\cong ...

**4**

votes

**1**answer

141 views

### Self intersection and deformations

Suppose I have a smooth projective surface $S$ and a curve $C\subset S$. The self-intersection of $C$ is by definition the degree of the restriction to $C$ of the normal bundle of $C$ inside $S$.
By ...

**6**

votes

**1**answer

230 views

### Etale fundamental of a parahoric group scheme

Let $p:X\rightarrow Y$ be a double cover of curves, denote by $$SU_n:=(p_*SL_n(\mathcal O_X))^{\tilde{\sigma}}$$
i.e. the $\tilde{\sigma}-$invariant part, the action of $\tilde{\sigma}$ is given by ...

**4**

votes

**0**answers

176 views

### Commutative algebraic version of algebraic geometric object

In my work, I have to understand certain objects in commutative algebra (for example Gorenstein rings, Cohen–Macaulay rings e.t.c). I have a reasonable background in commutative algebra (I suppose!) ...

**5**

votes

**1**answer

166 views

### are the coarse moduli schemes of finite etale covers of $\mathcal{M}_{1,1}$ smooth?

Let$\newcommand{\mM}{\mathcal{M}}$ $\mM_{1,1}$ be the moduli stack of elliptic curves. Let $R$ be a Dedekind domain, say $\mathbb{Z}[1/N]$ for simplicity, and suppose we have a finite etale cover:
...

**2**

votes

**1**answer

83 views

### Dimension of Quot scheme of zero dimensional quotients of a locally free sheaf

Given a locally free sheaf $E$ of rank $r$ on a (smooth, projective, algebraic) surface, I want to know the dimension of the scheme parametrizing the zero-dimensional (meaning they have zero ...

**57**

votes

**1**answer

2k views

### Is there a complex surface into which every Riemann surface embeds?

This question was previously asked on Math SE.
Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma ...