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

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2
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57 views

A conjecture like Cayley–Bacharach theorem

Let five conics and 12 points on a plane, one conic through six points, and one point lie on two conics. Then every cubic that passes through any eleven of the points also passes through the 12th ...
3
votes
0answers
57 views

Do very general hypersurfaces contain smooth surfaces with $c_1^2>2c_2?$

Let $X$ be a very general hypersurface of degree $d$ in $\mathbb{P}^n.$ Does $X$ contain a smooth surface $S$ with $c_1(T_S)^2>2c_2(T_S)$? For $d<<\sqrt{n}$ the answer is yes, as $X$ will ...
2
votes
0answers
55 views

Modules over an Azumaya algebra and modules over the associated Brauer-Severi variety

Assume $\mathcal{A}$ is an Azumaya algebra of rank $r^2$ on a smooth projective scheme $Y$ over $\mathbb{C}$. Let $f: X\rightarrow Y$ be the Brauer-Severi variety associated to $\mathcal{A}$. I read ...
10
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123 views

Del Pezzo surfaces and homotopy groups of spheres

A (complex) del Pezzo surface is a smooth projective complex surface with ample anticanonical line bundle. Such surface has a degree defined as the self intersection of the canonical divisor. It is ...
-2
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80 views

almost complex embedding of $S^2$ and $S^6$ into $\mathbb{C}^N$ [migrated]

In Which Spheres are Complex Manifolds? , I find that $S^2=\mathbb{C}P^1$ is a complex manifold and $S^6$ is an almost complex manifold. Are there references about: What is the smallest integer $N$ ...
3
votes
1answer
121 views

characteristic classes of symmetric product

Given a (real or almost complex) manifold $M$, Let the symmetric product be the quotient space $$ B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}_2 $$ where $$ \Delta=\{(m,m)\mid m\in M \} $$ and ...
2
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0answers
42 views

How can I include irreducibility in a Groebner basis calculation?

I'm trying to prove impossibility of certain systems of differential/polynomial equations using Groebner basis techniques. For example, consider the equation $qn = mf$, where each of the variables ...
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0answers
68 views

The self intersection class of exceptional divisor of 3-fold blown up along a curve

Suppose $X$ is a smooth complete variety of dimension $3$, let $\sigma\colon\widetilde{X}\to X$ the blow-up along smooth curve $C\subset X$, let $\sigma^{-1}(C)=E$ be the exceptional divisor, let $f$ ...
2
votes
1answer
86 views

Mestre-type algorithm for higher-genus curves?

Is there an analogous algorithm for genus $g>2$ curves that, given a complete set of invariants, outputs a curve with those invariants? (I'm interested in particular in $g=3$.) Any references ...
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47 views

What is the Segre class of a generating line of a cone

Suppose $U=\textrm{Proj}\ k[X,Y,Z,W]/(XY-Z^2)$ is the projective closure of an affine cone, let $V$ be a generating line of the cone $V=V(Y,Z)$, how do we calculate the Segre class $s(V,U)$? (We can ...
3
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140 views

Deformation of finite coverings between smooth projective varieties

Assume that we have a finite covering $$f \colon X \longrightarrow Y,$$ where $X$ and $Y$ are smooth, complex projective varieties of dimension $n$. Therefore we obtain a splitting $$f_* \mathscr{O}_X ...
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74 views

Chern classes of ideal sheaf of locally complete intersection

Let $C\varsubsetneq X$ be a reducible locally complete intersection closed pure $1$-dimensional subscheme in a smooth projective variety of dimension $3$ over a algebraically closed field $k$ of ...
2
votes
1answer
190 views

Indecomposable decomposition for a commutative ring

Let $R$ be a commutative ring with identity. We say that $R$ has an indecomposible decomposition if it can be wrighten as a finite direct sum of indecomposiable rings. Is there any characterization ...
7
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3answers
601 views

Category theory for Algebraic Geometry

How much of category theory should I know to view schemes, sheaves and cohomology concepts as concrete cases of abstract categorical concepts? Is there a textbook of category theory for AG people?
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87 views

Fundamental Group of SL_2 [on hold]

I am thinking whether there is a simple criterion or visible method to know the fundamental group of SL_2(R), or SL_2(F) with an arbitrary field F. Because SL_2(R) is already a 3-dimensional ...
3
votes
2answers
185 views

Connectedness of moduli of vector bundles

Let $X$ be a smooth projective variety. Given two vector bundles $V_1$ and $V_2$ such that $[V_1]=[V_2]\in K^0(X)$, can one expect that $V_1$ and $V_2$ can be connected by a family of vector bundles? ...
6
votes
1answer
426 views

A road to inter-universal Teichmuller theory

What would be a study path for someone in the level of Hartshorne's Algebraic Geometry to understand and study inter-universal Teichmuller (IUT) theory? I know that it heavily relies on anabelian ...
2
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0answers
85 views

Lie algebra of holomorphic vector fields

It's well known that the holomorphic vector fields on a complex manifold form a Lie algebra. In simplest situations, this Lie algebra can be described explicitly. For example, take $X=\mathbb{P}^n$, ...
2
votes
2answers
133 views

Counting number of $2\times 2$ unimodular matrices of particular type

From set of numbers from $\Bbb S=\{0,1,\dots,m\}$, how many distinct $3\times 3$ unimodular matrices parametrized by $(a,b,c,d,e,f)\in\Bbb S^6$ of following type can one form? \begin{bmatrix} a^2 ...
2
votes
0answers
81 views

An equality of discriminant and resultant divisors

Let $\Phi$ be the root system of a split group $G$ over a field $k$. The differentials $d\alpha$ of the roots define a polynomial called the discriminant $$\prod_{\alpha\in\Phi}d\alpha$$ on $\mathfrak ...
7
votes
1answer
275 views

About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've asked this question http://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text on math.stackexchange , however I don't think I will receive ...
3
votes
0answers
66 views

Unibranch partial normalization

In a paper I recently read something about the "unibranch partial normalization" of a curve. Say, $R$ is a local integral domain with maximal ideal $\mathfrak{m}$ and fraction field $K$. Is it ...
3
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0answers
109 views

Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2

Does every compact complex manifold of complex dimension greater than or equal two possess a nontrivial rank 2 holomorphic vector bundle?
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119 views

Derived global functions on (derived) stacks $BG$ and $G/G$

In Toen's Affine Stacks, he computes that $\mathcal{O}(B\mathbb{G}_a) = k[\epsilon]$ with $|\epsilon| = 1$ and trivial differential (where here $\mathcal{O}$ is computed in a derived sense, and we ...
0
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0answers
122 views

Quantities associated to deformed sheaves

I am trying to figure out what happens to "quantities" associated to a sheaf when one deforms it. I am actually interested in deforming a bounded complex of coherent sheaves but I want to make the ...
9
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0answers
229 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
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167 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
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0answers
116 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
0answers
90 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 ...
11
votes
3answers
583 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
2answers
180 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 ...
6
votes
1answer
185 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
1answer
302 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
0answers
121 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
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0answers
102 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
1answer
319 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) ...
8
votes
0answers
136 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 ...
3
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0answers
119 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
1answer
124 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
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0answers
143 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
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1answer
225 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 ...
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117 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
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173 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
1answer
121 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
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0answers
152 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
0answers
81 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
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1answer
115 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
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1answer
272 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
0answers
143 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 ...
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62 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 ...