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

**9**

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**0**answers

281 views

### Intrinsic definition of the weight filtration

Let $X$ be a smooth quasiprojective complex variety. Then Deligne (Theorie de Hodge II) defined a weight filtration on the Betti cohomology of $X$. The general philosophy is quite simple: express the ...

**5**

votes

**0**answers

178 views

### definition of “immersion” of schemes (without open or closed)

On Prop. 1.7 (a) on page 5 of Milne's Etale Cohomology book states:
Any immersion is quasi-finite.
A google search turned up definitions for "open immersion" and "closed immersion", never just ...

**3**

votes

**0**answers

117 views

### Commuting diagram, algebraic cycles and K-theory

What is the easiest way to see the veracity of the following commutative diagram?$$\require{AMScd}
\begin{CD}
K(X) \otimes K(Y) @>\text{ch}(-) \otimes \text{ch}(-)>> A(X, \mathbb{Q}) \otimes ...

**1**

vote

**0**answers

94 views

### Rank of the Jacobian of a family of hyperelliptic curves of genus 2

Assume tha $C$ be the hyperelliptic curve $y^2 = (x-a_1)\cdots (x-a_5)$ of genus $g=2$ and $a_i \in \mathbb{Z}$ and we know that the integers $a_i$ has the form $a_i= d_1^2 - d_i^2$ for some positive ...

**12**

votes

**3**answers

612 views

### Tangent space of Hilbert scheme

We have the following theorem:
Let $X$ be a projective scheme over an algebraically closed field $k$, and $Y \subset X$ a closed subscheme with Hilbert polynomial $P$. Then$$T_{[Y]}\text{Hilb}_P (X) ...

**4**

votes

**2**answers

182 views

### Vanishing natural transformation and strong generator

Let $X$ be a smooth projective variety (over the field of complex numbers). Let $T$ be strong generator of $D^b(X)$ : this means that every object in $D^b(X)$ can be obtained in a given finite number ...

**7**

votes

**1**answer

199 views

### Construction of coherent sheaf such that $\text{Proj}\,\text{Sym}\,(\mathcal{F}) = \text{Sym}^n X$

Let $X$ be a smooth projective curve. How do I construct a coherent sheaf $\mathcal{F}$ on $\text{Pic}^n X$ (i.e., the component of the Picard scheme of $X$ parametrizing line bundles of degree $n$) ...

**4**

votes

**1**answer

320 views

### Asymptotic $\int_M \mathrm{exp}[\mathbf{e}\left(n -\frac{t}{2\pi i}\right)] \left( 1 + \frac{5}{6} \mathbf{e}^2 \right)^{1/2} $ on quintic Calabi-Yau

Let $M = \{ G(x) = 0 \} \subseteq \mathbb{P}^4$ be a quintic Calabi-Yau and $\mathbf{e} \in H^2(M, \mathbb{Z})$ such that $\int_M \mathbf{e}^3 = 5$. Then as $t \gg 1$:
$$
\int_M
e^{n \mathbf{e}}
...

**3**

votes

**0**answers

128 views

### Does the Bombieri-Lang conjecture imply severe restrictions on rational points on twists of hyperelliptic curves?

According to Silverman, the Bombieri-Lang conjecture implies
that the rational points of surface on general type lie on
finite set of curves, except for a finite set of points.
Let $f$ be univariate ...

**3**

votes

**0**answers

105 views

### Elementary examples on sheaf extension

Let $V\subset\mathbb{P}^n$ be a projective variety and $C_V$ its conormal subvariety in $T^\ast\mathbb{P}^n$. Denote by $\mathscr{O}_{C_V}$ its structure sheaf, then when will the condition
...

**9**

votes

**1**answer

356 views

### Relationship between the syntomic cohomology of Kato and of Fontaine-Messing

Fix a prime $p$ and let $X$ be a $\mathbb{Z}_{p}$-scheme. Write $X_{n}:=X\otimes\mathbb{Z}/p^{n}$ and $\phi:X_{1}\rightarrow X_{1}$ for the absolute Frobenius. Let $X\hookrightarrow Z$ be a (suitable) ...

**12**

votes

**0**answers

141 views

### For all schemes w/Hilbert polynomial P, exists $m_P$ s.t. no higher cohomology, $I(k)$ generated by globally sections, multiplication is surjective

Consider the following theorem.
For every polynomial $P$, there exists an integer $m_P$ such that for all ideal subsheaves $I \subset \mathcal{O}_{\mathbb{P}^n}$ with Hilbert polynomial $P$ and ...

**4**

votes

**0**answers

148 views

### Pull-push formula?

There are many contexts in which the push-pull formula $f_*(f^*(\alpha)\cdot \beta) = \alpha \cdot f_*(\beta)$ holds. I am interesting mostly in the case of algebraic K-theory and Chow rings (under ...

**2**

votes

**1**answer

135 views

### Derived pullback of the coarse moduli morphism

Let $f: \mathcal{X}\to X$ be a morphism from a smooth DM-stack $\mathcal{X}$ to its coarse moduli space $X$. Assume that $X$ is also smooth. Is it true that $Lf^*$ is fully faithful and induces an ...

**4**

votes

**0**answers

147 views

### object in D^bCoh(P^2) with prescribed RHom's

Let $\mathbb{C}P^2$ denote the projective plane.
From reading the section of http://homepages.math.uic.edu/~coskun/gokova.pdf
which surveys Gieseker stable sheaves, I have understood that there are ...

**2**

votes

**1**answer

239 views

### When are direct products exact in the category of quasi-coherent sheaves?

(This question is crossposted from MSE, since there the question did not recieve any attention whatsoever.)
I would like to know if there is a description (or at least some sufficient condition ...

**1**

vote

**0**answers

121 views

### Reference for the statement that the complement of an affine open has codimension one

The following statement seems to be "well-known", but I am unable to find a reference in the standard literature. Could someone suggest a reference?
Let $X$ be a separated normal connected Noetherian ...

**2**

votes

**0**answers

175 views

### anabelian analogues of the Weil Pairing?

Classically the Weil pairing for an abelian variety $A$ over a field $k$ (say of characteristic 0 for simplicity) can be thought of as an (especially nice) morphism of galois modules:
...

**2**

votes

**1**answer

124 views

### The cohomology ring of a compact toric manifold

Given a compact toric manifold $X$ with Picard number $r$ it is well known that the cohomology ring $H^*(X;\mathbb{C})$ is a quotient of the polynomial ring $\mathbb{C}[p_1,\dots,p_r]$ by the ideal ...

**2**

votes

**0**answers

159 views

### Morphism of Shimura varieties and differential equations

Is there a way of constructing a morphism between Shimura varieties using differential equations? Maybe, this looks like a completely ridiculous question, so I think that I should explain the context ...

**3**

votes

**0**answers

88 views

### Combinatorial interpretation for a toric intersection number

Let $X$ be an $n$-dimensional toric variety and let $D$ be an effective divisor (eg nef or ample). Is there a combinatorial interpretation (eg in terms of the fan or polytope) of the intersection ...

**2**

votes

**1**answer

120 views

### Definition of Strongly Stable 0-cycle

I am not sure whether this question deserves to be asked in this forum, but I have no other choice as I can't find the definition anywhere. So here is the question:
When is a 0-cycle on $\mathbb P^n$ ...

**3**

votes

**1**answer

275 views

### Serre duality over a non-algebraically closed field

Suppose $X$ is a projective smooth variety over a non-algebraically closed field , do we still have $Ext^i(F,\omega)\to H^{n-i}(X,F)^{\vee}$? (Hartshorne's proof Thm III 7.6 requires $k$ to be ...

**2**

votes

**0**answers

150 views

### Rank of the Jacobian of twists of hyperelliptic curves

Suppose that a hyperelliptic curve $C$ of genus $g \geq 4$ is given by the equation
$$\displaystyle C: y^2 = a_0 x^{2g+2} + a_1 x^{2g+1} + \cdots + a_{2g+2} = f(x).$$
The Jacobian variety $J(C)$ of ...

**1**

vote

**0**answers

72 views

### Is the elementary transformation along a curve decomposable?

Let $S$ be a surface. Let $L$ be an ample line bundle on $S$. Let $C\in |L|$ be a curve on $S$, and let $A$ be a globally generated line bundle on $C$ of degree $d$ and with 2 sections.
Then we get ...

**0**

votes

**1**answer

120 views

### perfect modules over polynomial algebra

This may be obvious. My question is short:
$R$ is the polynomial algebra $\mathbb{k}[X_{1},\dots , X_{n}]$. Is the $R$-module $\mathbb{k}$ perfect in the sense that $\mathbb{k}$ is a compact object ...

**1**

vote

**0**answers

240 views

### Generic triviality of $G$-bundles

Let $k$ be an algebraically closed field and $X$ a curve over $k$. Then any $G$-bundle on a curve (where $G$ is reductive and connected) is generically trivial. This is the one of the main results of ...

**5**

votes

**1**answer

299 views

### Is there a geometric proof for the upper semicontinuity of fiber dimension in algebraic geometry?

One of the first theorems encountered in algebraic geometry is the upper semicontinuity of fiber dimension:
Let $ f : X \to Y $ be a surjective regular map between irreducible varieties with ...

**6**

votes

**1**answer

253 views

### Is there a non-smooth algebraic group scheme in char $p$, all of whose defining relations have degree less than $p$?

Let $k$ be an algebraically closed field of characteristic $p>0$.
All the examples of non-smooth algebraic group schemes over $k$ that
I have seen (apart from "artificial" examples; see below) have ...

**7**

votes

**1**answer

254 views

### Definition of p-adic modular forms

I have been reading Hida's book "p-Adic automorphism forms on Shimura varieties" and I don't understand a point.
He first describes p-adic modular forms of tame level N as functions on the Igusa ...

**4**

votes

**0**answers

117 views

### Proof of theorem of Nagata, modify step for nonzero divisor?

Here is the theorem of Nagata I am working with.
Let $G$ be a geometrically reductive group acting rationally on a finitely generated $k$-algebra $R$. Then the ring of invariants $R^G$ is finitely ...

**3**

votes

**1**answer

251 views

### Ramification divisor on curves in weighted projective space

I was hesitant about posting this question here, but since it deals with a partially unanswered question already on this site I figured that this would be the best place for it. I apologise in advance ...

**2**

votes

**0**answers

99 views

### Full exceptional collections in derived category of coherent sheaves on non-compact varieties

Let $X$ be a smooth algebraic variety over $\mathbb{C}$, and $D^b(X)$ its bounded derived category of coherent sheaves. Then a full exceptional collection will lead to significant simplifications in ...

**12**

votes

**0**answers

355 views

### Easiest way to see Theorem 1.2 of Deligne-Mumford's “The irreducibility of the space of curves of given genus”?

Theorem 1.2 of Deligne-Mumford's 1969 IHÉS paper, "The irreducibility of the space of curves of given genus," is as follows.
If $g \ge 2$ and $C$ is a stable curve of genus $g$ over an ...

**0**

votes

**0**answers

55 views

### Characterization of Singular locus

Let A be a complete regular local ring over a field k and B be a complete normal local ring over a field k. We assume that (Krull-dimension of A) > 1.
We consider the ring homomorphism f: A ---> B, ...

**3**

votes

**1**answer

147 views

### What is the type of the surfaces $x^5 - y^5 + z^2 + x=0$ and $x^5 - y^5 + z^2 + x+1=0$?

Crossposted from MSE.
I am interested what is the type of the surfaces over the
rationals
$$ x^5 - y^5 + z^2 + x=0$$
and
$$ x^5 - y^5 + z^2 + x+1=0$$
Magma's ...

**5**

votes

**1**answer

195 views

### Hyperelliptic curves with fixed genus and many rational points

It is a famous theorem of Faltings, previously a conjecture by Mordell, that any algebraic curve of genus at least $2$ defined over the rational numbers have at most finitely many rational points. A ...

**2**

votes

**1**answer

110 views

### The line bundle of the divisor at infinity of the moduli stack of stable curves of genus $g \ge 2$

Let $\overline{\mathscr{M}}_g$ be the $\mathbb{Z}$-algebraic stack of stable curves of genus $g \ge 2$, as constructed in the paper of Deligne and Mumford. The degeneracy locus of the universal stable ...

**5**

votes

**0**answers

134 views

### Simplicity of a rank 2 vector bundle over a principally polarized abelian surface

Let $A := \textrm{Jac}(C)$ be the Jacobian of a genus $2$ curve $C$, with principal polarization $\Theta$.
Studying some branched covers of $A$, I was led to consider rank $2$ holomorphic vector ...

**0**

votes

**0**answers

76 views

### A property of the semi-local ring of the normalization of a singular curve

I have two following questions.
1) Let $R$ be a local ring in an algebraic function field of one variable over an algebraic closed field $k$. Let $\bar{R}$ and $m$ be its integral closure and maximal ...

**3**

votes

**2**answers

167 views

### Examples of (non-normal) unibranched rings?

For a local integral domain $R$ the following are equivalent:
a) The integral closure of $R$ in its fraction field (i.e., the normalization of $R$) is again local.
b) The henselization of $R$ is ...

**4**

votes

**1**answer

305 views

### Training towards research on k3 surfaces

I am a graduate student learning basic algebraic geometry (from Hartshorne, Shafarevich). I'm planning to work in k3 surfaces (arithmetic and geometric properties, in my guide's words). I came to know ...

**3**

votes

**0**answers

58 views

### Families of trigonal curves with hyperelliptic limit

Suppose I have a family of trigonal curves $C\to D$ over a closed disk $D$ where the central fiber $C_0$ is hyperelliptic (this is of course possible since the hyperelliptic locus is in the closure of ...

**1**

vote

**0**answers

121 views

### Isomorphism classes of curves $x^{m}+y^{n}=constant$

Let us fix two coprime integers $m,n\geq2$ and denote by $K$ the field $K=\mathbb{C}(t)$ of rational functions in one variable over the complex numbers. Consider, for every nonzero constant $c\in ...

**1**

vote

**0**answers

98 views

### How to check with a CAS if a surface is of general type?

The main question is:
How to check with a CAS if a surface is of general type?
Magma's function KodairaEnriquesType is close to this,
but doesn't always work.
...

**7**

votes

**1**answer

163 views

### Smooth quadric hypersurface, Hilbert scheme is blowup of Grassmannian?

Let $Q \subset \mathbb{P}^n$ be a smooth quadric hypersurface. Where can I find a proof of/can anyone supply a proof of$$\text{Hilb}_{2m + 1}(Q) \cong \text{Bl}_{OG(3, n+1)}G(3, n+1)?$$Can we conclude ...

**1**

vote

**1**answer

124 views

### Socle of Almost Complete Intersections

Let $(A,m)$ be a complete Artinian local ring over a field $K$.
We focus on almost complete intersection ring $A$ of the form
$A = K[[X_1,...,X_N]]/(f_1,...,f_{N+1})$.
We assume that none of $f_i$ ...

**3**

votes

**2**answers

268 views

### Is it normal surface of general type to have infinitely many positive rank elliptic curves?

Cross-posted from MSE.
I am not good at algebraic geometry and almost surely am
misunderstanding something.
Got an alleged argument against Bombieri-Lang conjecture and
would like to know what the ...

**1**

vote

**1**answer

89 views

### How to define the internal hom between presheaves valued in cotensored categories?

First let $\mathcal{V}$ be a closed symmetric monoidal category
and $\mathcal{M}$ be a category enriched over $\mathcal{V}$. Moreover we assume $\mathcal{M}$ is cotensored, or powered over ...

**1**

vote

**0**answers

114 views

### addition on an affine scheme [closed]

At the Brenner's introduction to the geometric view of the tight closure the author states that an affine scheme has a natural addition (this addition will be extended to the vector bundles). I wonder ...