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

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117 views

### Learning roadmap in Algebra [on hold]

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas:
a) Commutative Algebra
b) Field Theory and Galois Theory
c) Homological Algebra
My question is ...

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**0**answers

37 views

### u-Invariants of p-adic function fields

In his Paper "Fields of u-invariant 9" Oleg Izhboldin points out that for a algebraic closed, finitely generated field $k$ we have $u(k)= 2^{cd(k)}$. In particular we have
...

**3**

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78 views

### Singularities in mixed characteristic

Let $R$ be a regular local ring in mixed characteristic. Moreover, I assume that $R$ is the local ring of a point on a smooth $\mathbb Z_p$-scheme and that $R/pR$ is regular. ($\mathbb Z_p$ is the ...

**3**

votes

**1**answer

98 views

### Compositional inversion and generating functions in algebraic geometry

The exponential generating function of the graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus zero satisfying the associativity equations of physics (the WDVV ...

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**0**answers

40 views

### Algebraic approach to showing trigonometric equations have no solution

I have very little background in algebra and algebraic geometry, so please bear with me.
I am trying to show that certain systems of trigonometric polynomial equations generally have no solution. One ...

**4**

votes

**1**answer

101 views

### p-adic L-function of curves

Given a smooth projective curve $C$ over $\mathbb{Q}$ one has the $L$-function $L(C, s)$ and the Beilinson conjectures predict its values at integers $s=n$ in terms of regulators.
Is there a p-adic ...

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votes

**2**answers

173 views

### Sheaf of isogenies representable?

It is well-known that "the" stack of elliptic curves (allow me to be vague as to singular curves, compactifications etc) has a presentation by a groupoid in schemes. One of the things that needs to be ...

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vote

**1**answer

87 views

### Compute higher direct image for Gm under open embedding

Let $U \subset \mathbb P^1$ be an open subset of projective line (over $\mathbb C$) after removing $r$ points and $j: U\hookrightarrow \mathbb P^1$ an open immersion. How do I compute $R^1j_*\mathbb ...

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**1**answer

161 views

### transcendence of beta values

(1) Can anybody suggest a readable reference for Schneider's theorem that the number
$$
\beta(a, b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
$$ is transcendental for $a, b \in \mathbb{Q}$ such that none ...

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62 views

### plane curves with two points of high multiplicity

Let $\mathcal{C}$ be an irreducible plane curve in $\mathbb{P}^2_\mathbb{C}$ of degree $d$. Let $D$ be a quartic with three irreducible components with normal crossing singularities, i.e. a conic and ...

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votes

**2**answers

140 views

### Contracting a rational curve in a Calabi-Yau threeolfd

Let $X$ be a Calabi-Yau threefold and $C \subset X$ be a rational curve with $N_{C/X}\cong \mathcal{O}\oplus \mathcal{O}(-2)$. Can one contract the curve $C$? Assuming the answer is yes, what kind of ...

**17**

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**5**answers

658 views

### Conceptual algebraic proof that Grassmannian is closed in Plucker embedding

I'm planning lectures for my intro algebraic geometry course, and I noted something awkward that is coming up. We're starting projective varieties soon. Of course, we'll prove that projective maps are ...

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**0**answers

118 views

### Locally Closed Orbits in Real Algebraic Geometry

Let $G$ be a real algebraic group, and let $X$ be a real affine $G$-variety. I am looking for conditions on $G$ and $X$ for which the $G$-orbits are known to be locally closed in the Zariski topology ...

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votes

**1**answer

115 views

### deformations of vector bundles on curves

Let $X$ be a smooth algebraic curve. Suppose I have a flat family $V_y\to X$ of vector bundles on $X$ over an affine scheme $S$. Let $p=Spec(k)$ be one geometric point of $S$. If the determinant of ...

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votes

**1**answer

79 views

### Is the support of two odd theta characteristics on a generic curve disjoint?

Concise version of the question
On a generic curve of genus $g$, the odd theta characteristics will have exactly one global section. Therefore each odd theta characteristic will correspond to a ...

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39 views

### Localization on orbit type submanifolds (generalization of Atiyah-Bott-Berline-Vergne)

In equivariant cohomology, the Atiyah-Bott-Berline-Vergne localization theorem says roughly speaking that the integral of an equivariant cohomology class on the $G$-manifold $M$ has only contributions ...

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101 views

### path integral and index theorem

I actually have an integral which is used to prove Atiyah-Singer index theorem for spin complex in a path integral fashion. The integral I need to evaluate is following (in simplified form)
$\int ...

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**0**answers

37 views

### Connection between Strebel differentials, ribbon graphs, and Belyi maps

In this paper, a nice story is woven regarding the connection between quadratic differentials on Riemann surfaces, so-called 'ribbon graphs' drawn on those surfaces, and Belyi maps. However, I am ...

**-1**

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47 views

### An isogeny from a split algebraic torus [migrated]

Suppose that there is an isogeny (in the category of commutative algebraic groups) from a split algebraic torus to a semi-abelian variety. Does it follows that this semi-abelian variety is also an ...

**2**

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**0**answers

58 views

### Explicit equations for conormal bundle to an affine toric variety

Let $L \subset \mathbb{Z}^n$ be a lattice and let $X_L$ be the closed toric subvariety of $\mathbb{C}^n$ cut out by the lattice ideal $I_L = \{x^{l_+} - x^{l_-} \,| \, l_+, l_- \in \mathbb{N}^n \text{ ...

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77 views

### preservation of localness among certain Krull domains

The following question essentially appeared (http://math.stackexchange.com/questions/931801/preservation-of-localness-among-certain-krull-domains) on math.SE a while ago, but nobody has done anything ...

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**0**answers

117 views

### Containment of two varieties with a lot of intersection [migrated]

Given a projective variety $X\subset \mathbb P^n$ and a curve $C\subset \mathbb P^n$, when can I conclude that $C\subset X$, from the fact that $C$ and $X$ have 'many' points in common. I.e., is there ...

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**1**answer

207 views

### Can the Grothendieck ring of varities over a field $k$ be defined for non separated schemes?

The Grothendieck ring of varieties over a field $k$ is the abelian group generated by isomorphim classes $[X]$ of separated, reduced $k$-schemes $X$ of finite type with the relation
$[X]=[Y] + ...

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**1**answer

182 views

### Decomposition of symmetric homogeneous polynomials

Can every symmetric polynomial of degree $r$ in $d$ variables that has no constant term be written as a sum of the $r$th powers of linear polynomials in $d$ variables and a homogeneous polynomial of ...

**-1**

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**0**answers

62 views

### Isogeny of abelian varieties over general fields [on hold]

We know that given an abelian variety $X$ over an algebraically closed field $K$ of characteristic $0$ and any integer $n$ the induced map $[n]:X \to X$ is an isogeny. As far as I understand this ...

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63 views

### On the universal property of certain representable functors and rational sections

Let $P_1,P_2$ be two Hilbert polynomials of subschemes in $\mathbb{P}^n$. Denote by $H_{P_1,P_2}$ the corresponding flag Hilbert scheme (parametrizing pairs $(X\subset Y)$ where $X$ has Hilbert ...

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**1**answer

182 views

### Different notions of convergence of complex subvarieties

Let $X$ be a smooth complex algebraic variety (or, better, complex analytic manifold). Let $\{C_i\}$ be a sequence of compact algebraic subvarieties (resp. analytic reduced subspaces) which converges ...

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**1**answer

115 views

### An application of the Grauert's upper semi-continuity theorem

Let $X$ be a smooth projective variety, $A$ a complete discrete valuation ring, $Y=\mbox{Spec} A$ and $f:X \to Y$ a smooth, projective, surjective morphism. Denote by $y$ the closed point of $Y$. Let ...

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**0**answers

211 views

### Singularities arising from the Minimal Model Program (an algebraic point of view)

I will start the story by the end:
Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ...

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92 views

### On the Picard group of a product of projective varieties [on hold]

Let $K$ be a field of characteristic zero, $X$ a smooth projective curve on $K$ and $Y$ a Fano variety over $K$. Consider the natural projection morphism $\mbox{pr}_1$ (resp. $\mbox{pr}_2$) from $X ...

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**1**answer

208 views

### Groups and pregeometries

Definition.
For an infinite structure $\mathcal{A}$ and $cl : P(dom(\mathcal{A})) \longrightarrow P(dom(\mathcal{A}))$ , we say
that $(\mathcal{A}, cl)$ is a structure carrying an $\omega$-homogeneous ...

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votes

**1**answer

402 views

### Is it easy to prove that $\sum_n |X(\mathbb{F}_{q^n})| t^n$ is rational?

Background: Let $X$ be an algebraic variety over a finite field $\mathbb{F}_q$. One of the successes of Etale cohomology - previously achieved by Dwork- was proving the rationality of the Zeta ...

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118 views

### When is there a polynomial transformation? [closed]

First part: given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}=\frac{P_3(f(x_1,x_2,\dots,x_n))}{P_4(f(x_1,x_2,\dots,x_n))}|\det (J(f(x_1,x_2,\dots,x_n)))|$$ where $P_i$ is polynomial ( that ...

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179 views

### Are these two “FUNCTORS” adjoint?

I am considering the following correspondence:
Let $X$ be quasi compact quasi separated schemes.Consider a pseudo functor \begin{equation}Sch\rightarrow CAT :U\mapsto Qcoh(U),f:U\rightarrow V\mapsto ...

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110 views

### If the direct image of f preserves coherent sheaves on notherian schemes,how to show f is proper?

The other direction is well known
I think it is true and I was told by several other guys doing algebraic geometry that it is indeed true but they did not know how to prove.I am also wondering whether ...

**1**

vote

**1**answer

187 views

### Blow-ups and cohomology

I'm trying to understand how to compute the Chow ring of a blow-up.
Let $W\subset \mathbb P^4$ be a smooth surface and let $X$ be the blow-up of $\mathbb P^4$ along $W$ with exceptional divisor $E$. ...

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votes

**1**answer

227 views

### What are the exact holomorphic Lagrangians in complex 2-space?

In an exact symplectic manifold, i.e. where the symplectic form can be written $\omega = d \lambda$, it's natural to look for exact Lagrangians, i.e. $L$ on which $\lambda_L = df$. One reason is ...

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**0**answers

157 views

### Tannaka categories and reductive groups

The group associated to a Tannaka category $T$ over a field is pro-reductive if and only if $T$ is semi-simple.
Pro-reductive groups make sense over any scheme.
Is there an extension of the theory ...

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**1**answer

118 views

### Categorical characterization of closed imbeddings

Let $f\colon X\to Y$ be a morphism of schemes.
Let $F_X$ and $F_Y$ be the contravariant functors from the category $Sch$ of schemes to the category of sets defined via the Yoneda construction, i.e. ...

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**0**answers

92 views

### Universal covering space of a Zariski open subset of projective space

Let $U$ be a Zariski open subset of $\mathbb P^n_{\mathbb C}$. Assume $U$ is the complement of some divisors.
Have the possible universal covering spaces of $U$ been classified?
Do we know when the ...

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105 views

### Is there such thing as the Gorensteinification of a one-dimensional local ring?

That is, given $A$ local, reduced and one-dimensional, is there a finite extension $A\to B$ where $B$ is Gorenstein?

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**1**answer

234 views

### The space of varieties between two given varieties

Let $\mathbf{P} = \mathbf{P}^n(k)$ be the $n$-dimensional projective space over a field $k$, let $A, B$ be projective varieties in $\mathbf{P}$ such that $A \subset B$. Now define
$V(A,B)$ to be the ...

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186 views

### Reference request: Beilinson-Bernstein for finite-dimensional reps and category O

I think I’ve once been told that under the Beilinson-Bernstein correspondence, finite-dimensional representations of a semisimple Lie algebra $\mathfrak{g}$ correspond to (twisted) D-modules on $G/B$ ...

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121 views

### which sections of elliptic curves are conjugate?

Suppose you have a relative elliptic curves $f : E\rightarrow S$ (say $S$ is connected). Then suppose you have two sections $g,g' : S\rightarrow E$, corresponding to two sections $g_*,g'_*$ to the map ...

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**1**answer

132 views

### Pulling back quasi-coherent sheaves from a quotient stack

In a problem I am trying to solve, the following situation occurs. $X$ is a smooth variety and $G$ is a reductive group acting transitively on $X$. We have the stack $X/G$ and a morphism $\pi : X \to ...

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59 views

### stability notion of nets of quadrics

A net of quadrics in $\mathbb{P}^n$ is a plane in $\mathbb{P}^N$, where $N=\frac{n(n+3)}{2}$. So the space of net of quadrics is the Grassmannian $Gr(3,N+1)$. The group $SL_{n+1}(\mathbb{C})$ acts on ...

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103 views

### infinite dimensional germs of schemes and tangent spaces

(The question of the type "how to define?")
Let $(R,\mathfrak{m})$ be a local ring over a field $k$ of zero characteristic. Consider the matrices over this ring, $Mat(m,R)$. I think of $Mat(m,R)$ as ...

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**1**answer

270 views

### Does there exist a Fano variety with torsion in $H^3$?

Let $X$ be a (smooth) Fano variety over $\mathbb{C}$. If $\dim(X)=3$, inspection of the Iskovskikh-Mori-Mukai lists seems to indicate that $H^3(X,\mathbb{Z})$ is torsion free. Is there a theoretical ...

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**1**answer

129 views

### Hasse principle and twists of $\mathbb{P}^n$ [closed]

Let $X$ be a twist of the $n$-th projective space, seen as a $K$-variety for some number field $K$. For $n = 1$, the Hasse principle holds for $X$.
My question is: for which $n >1$ does the ...

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84 views

### The meaning of induced sheaf $\mathscr F_y$ in Hartshorne's Corollary III.9.4

I do not quite understand Corollary III.9.4 on page 255 of Hartshorne's Algebraic geometry.
I quote the corollary here before I post my questions:
Let $f:\, X \to Y$ be a separated morphism of ...