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

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6
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0answers
32 views

Why should intersection cohomology and quantum cohomology be related for a symplectic resolution?

In http://arxiv.org/pdf/1410.6240.pdf M. McBreen and N. Proudfoot conjectured a precise relationship between the quantum cohomology of a symplectic resolution and the intersection cohomology of the ...
1
vote
0answers
48 views

Calabi-Yau theorem on Arithmetic Variety

Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$. Let $\omega$ be a Kaehler current of $\mathcal X(\mathbb C)$. ...
1
vote
0answers
43 views

A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$ where $a,b$ are two given points in the plane and $\lambda$ is a constant. Now we consider the ...
2
votes
0answers
57 views

Lefschetz on étale fundamental group for quasi-projective varieties

If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$ $$\pi^1(H)\to\pi^1(X)$$ is an isomorphism, ...
-3
votes
0answers
39 views

Negative Arithmetic Line bundle [on hold]

Let $X$ be an arithmetic variety and $L$ be an Arithmetic Line bundle, then how can we define a negative Arithmetic line bundle ?
2
votes
0answers
90 views

Deligne-Simpson problem for classical groups

Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation $$ A_1 +...+A_n ...
2
votes
0answers
79 views

Degree formalism for line bundles on Deligne-Mumford stacks

Let $k$ be an algebraically closed field and let $\mathcal{C}$ be proper, Cohen-Macaulay, purely $1$-dimensional Deligne-Mumford stack over $k$. From looking at section 4.3 on page 135 of the paper ...
2
votes
0answers
57 views

functor from complex algebraic variety to constructible function

I asked the same question on mathstackexchange but didn't get any response so I guess I should ask here. If you think it is a duplicate and it is not appropriate to post it here, I will delete this ...
0
votes
0answers
56 views

Cone of curves in blow-ups

Let $X_s = Bl_{L_1,...,L_s}\mathbb{P}^3$ be the blow-up of $\mathbb{P}^3$ in $s$ general lines $L_1,...,L_s$. The exceptional divisor $E_i$ over $L_i$ is isomorphic to ...
1
vote
0answers
51 views

Smooth points of the secant variety with a given tangent space [on hold]

Let $X\subseteq\mathbb{P}^{N}$ be an algebraic variety of dimension $n$. Let $(x,y)\in X\times X-\Delta_{X}$ and $z\in\langle x,y\rangle\subseteq SX$, where $SX$ is the secant variety of $X$. I want ...
0
votes
0answers
31 views

Open Hamiltonian Gromov-Witten Invariants

Both open Gromov-Witten invariants and Hamiltonian Gromov-Witten invariants have been studied. I am interested in knowing whether anyone has considered open Hamiltonian Gromov-Witten invariants ...
2
votes
0answers
121 views

splitting property of etale covering

Theorem (Global Splitting): Let $X$ be an integral separated normal scheme flat and of finite type over $\mathbb Z$. Let $\phi: Y\rightarrow X$ be a connected etale covering which splits completely ...
4
votes
0answers
115 views

Flat + locally of finite presentation + monomorphism = open immersion

It is known that the following are equivalent for an epimorphism $A \to B$ in $\mathbf{CRing}$: Let $S$ be the set of elements $a \in A$ such that $A [a^{-1}] \to B [a^{-1}]$ is an isomorphism. Then ...
3
votes
0answers
317 views

Grothendieck, A Place to Begin [on hold]

I'm finishing up an undergrad degree in mathematics and am beginning to think about areas of research. I know that the work of Grothendieck is considered the cornerstone of modern algebraic geometry, ...
-2
votes
0answers
73 views

A question about Kähler Einstein metric [on hold]

Let $X$, and $Y$ are Kähler manifolds and $f:X\to Y$ is birational and let on $(Y,\omega)$ we have $\text{Ric}(\omega)=-\omega$, then Kähler Einstein metric on $X$ can be of which form? can we say it ...
1
vote
0answers
23 views

Common Point of Intersection of n-dimensional ellipsoids [on hold]

Suppose we have two ellipses in 2-dimensions centered at the origin. It is easy to visualize that (unless one is contained in the other) they will have 4 points of intersection. Can we say that in ...
4
votes
0answers
95 views

Moduli space of complex Tori [duplicate]

Is there any explicit computation for the Weil-Petersson metric on moduli space of Tori of complex dimension n?
8
votes
1answer
178 views

Generalization of the rigidity lemma in birational geometry

Let $X,Y,Z$ be projective varieties, and let $f:X\rightarrow Y$, $g:X\rightarrow Z$ be dominant morphisms. Assume that all the fibers of $g$ have the same dimension and are connected. If there exists ...
2
votes
0answers
71 views

extending local systems on a neighbourhood

Let $Y$ an affine finite type scheme over an algebraically closed field $k$. Let $S$ be a closed subscheme of $Y$ and $Y'$ the henselization of $Y$ along $S$. If we have a $\mathbb{Z}_{\ell}$ local ...
1
vote
0answers
42 views

moduli space of curves under prescribed tangency conditons

We consider an irreducible component of the Hilbert Scheme of curves in $\mathbb P^2$. Denote it as $\mathcal D.$ We fix a line $L$ and a point $A\in L.$ Denote $\mathcal D_0$ as the subscheme of ...
1
vote
1answer
125 views

Direct image for crystals?

If we get a morphism $f : X \to Y$ of schemes over $k$, how should I define the direct image functor $$ f_* : Crys(X) \to Crys(Y)? $$ By a crystal I mean a quasi-coherent sheaf $M$ on $X$ with a ...
1
vote
1answer
129 views

l-adic local system. on hensel schemes

Let $k$ be a field, $\ell$ a prime different from the characteristic. If I take $S$ a closed subscheme of $Y$, which is a $k$-scheme of finite type, is it true that any $\mathbb{Z}_{\ell}$-local ...
2
votes
0answers
98 views

Moduli space of log Calabi-Yau varieties exists?

Let $\mathcal M$ be a moduli space of pair varieties $(X,D)$ which $K_X+D$ is trivial and $D$ is a divisor on Kahler variety $X$. I am looking for a proof that such moduli space exists? The log ...
11
votes
0answers
296 views
+200

Unifying (& “justifying”) the various definitions for differential operators

Reading about differential operators in different sources I've picked up several definitions which are not obviously equivalent (to me). Here they are: Definition 1 ("naive"): Let $X$ be a (real) ...
1
vote
0answers
49 views

Pairing for non-uniformizable Anderson T-motives

Let $M$ be an Anderson T-motive (the simplest case, i.e. abelian in the meaning of [G] Goss, Basic structures of function field arithmetic, Def. 5.4.12, over $A^1$, having $N=0$), and let $H_1(E)$ ...
1
vote
1answer
231 views

Reference request for an introduction to deformation theory in algebraic geometry

I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through ...
2
votes
0answers
89 views

DG natural transformation Serre functors

This question might be really easy (or stupid), but I have only vague (heard-about) knowledge of DG categories, so I don't know where to look for an answer. Let $X$ be a smooth projective variety ...
10
votes
1answer
288 views
+50

Obstructed automorphisms of schemes

Let $X$ be a smooth projective scheme over a field $\mathbf{k}$ of characteristic zero such that $\mathrm{H}^0(X, \mathrm{T}X)$ vanishes, and let $f$ be an automorphism of $X$. I would like to have an ...
2
votes
0answers
91 views

Open nature of $\mathcal{H}om$ functor/upper semi-continuity of $\operatorname{Ext}^i$

Let $k$ be an algebraically closed field, $T$ a $k$-scheme (can assume connected) and $X$ a projective variety over $k$. Let $\mathcal{F}$ be a coherent (pure) sheaf on $X \times_k T$ flat over $T$. ...
14
votes
1answer
392 views

Algebraic spaces as locally ringed spaces

Let $S$ be a scheme (although I am more than happy to have $S=\text{Spec}(k)$ for a field $k$) and $\mathsf{AlgSp}/S$ the category of algebraic spaces over $S$. Does there exist an embedding ...
4
votes
1answer
451 views

Algebraic Geometry needed for Kähler-Einstein metric

I am a Master's student interested in Differential Geometry / Geometric Analysis. Currently active research is going on in Kähler-Einstein / Extremal Kähler metric. I was wondering how much Algebraic ...
5
votes
1answer
219 views

Symplectic orthogonality and projective duality: how do they work together?

Let $(V,\omega)$ be a $2n$-dimensional linear symplectic space, and $(\mathbb{P}V,\theta_\omega)$ the corresponding $(2n-1)$-dimensional contact manifold. Given a smooth $(n-1)$-dimensional smooth ...
3
votes
0answers
184 views

Roadmap for the ideas expressed in Grothendieck's Esquisse d'un Programme

I would like to understand Grothendieck's Esquisse d'un Programme more. Are there any references that would help me, and are there modern works pursuing the same themes? At this point I am still ...
-1
votes
1answer
152 views

Distinguished triangle and short exact sequence [on hold]

Forgive me for asking an elementary question. Given coherent sheaves $A$, $B$, $C$ and morphisms $B\xrightarrow{f} C\xrightarrow{g} A$ which give rise to the distinguished triangle $A[-1] \rightarrow ...
2
votes
0answers
127 views

Computing intersection number of two arithmetic line bundles

I have some questions in Arithmetic Arakelov geometry Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$ and ...
2
votes
1answer
95 views

(Partial) crepant resolutions

Consider de orbifold $\mathbb{C}^2$/$\mathbb{Z}_n$. In this case a full crepant resolution exists and it is unique. However, this orbifold admits partial resolutions. So my question is: Are all those ...
5
votes
0answers
121 views

Comparison of sheaves of modular forms

Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$: $e^*\Omega^1_{E/X}$ and ...
4
votes
0answers
68 views

Non-universally trivial Chow group of zero-cycles on Fano hypersurfaces

Let $X$ be a smooth projective variety over a field $k$. By (one) definition, the Chow group of zero-cycles $CH_0(X)$ is universally trivial if, for every field extension $k \subset K$, the degree map ...
1
vote
0answers
97 views

Conjugacy scheme, fppf versus GIT

I would be glad to have some guidance in the following. Let $k$ be an algebraically closed field. Let $G$ be a connected reductive group over $k$. Denote by $\mathfrak{c}$ the Zariski spectrum of the ...
3
votes
0answers
80 views

henselizations along closed subscheme

Where can I find some references about henselizations ablong a closed subscheme? For example if I take a map $Y\times\mathbb{A}^{1}\rightarrow Y$ and $Z$ a closed subscheme. Let $Y_{Z}^{h}$ the ...
1
vote
0answers
73 views

Fiber of the specialization map of Picard groups

Let $R$ be a Henselian discrete valuation ring with residue field $k$ of positive characteristic and fraction field $K$ of characteristic zero. Let $\pi:X_R \to \mathrm{Spec}(R)$ be flat, projective ...
4
votes
3answers
156 views

Minimal “subset” of a set of homogeneous polynomials with same solution space

Suppose $A:=\{f_1,\dots,f_m\}\subset \mathbb{C}[x_1,\dots,x_n]$ with $m>n$ is a set of homogeneous polynomials of equal degree $d>0$. Suppose further that the variety they define consists of a ...
-3
votes
0answers
34 views

equation for triangular membership funtion [closed]

I am working on fuzzy logic. Although I know the equations for triangular membership function but I can't figure out how they are derived.Are they derived from slope of line concept or some other ...
6
votes
0answers
262 views

Interuniversal Teichmuller theory's applications

Apart from a proof of the ABC conjecture -and its accepted consequences- are there applications of Mochizuki's IUT? In particular are there already widely accepted applications? Does it shed ...
6
votes
2answers
313 views

First Galois cohomology of Weil restriction of $\mathbb{G}_m$

Let $L/K$ be a finite Galois extension, write $G:= Gal(L/K)$. Denote by $R = Res(\mathbb{G}_m)$ the Weil restriction of $\mathbb{G}_m$, from $L$ to $K$. I want to show that its first Galois cohomology ...
2
votes
0answers
85 views

Parametrizing binary quartic forms with the kernel of obstruction map

This is my first post, so I apologize beforehand if my questions are too elementary for this site. In this paper by Fisher https://www.dpmms.cam.ac.uk/~taf1000/papers/testeqtc.pdf it is explained ...
2
votes
1answer
95 views

What finite groups are stabilizers in Kirwan's desingularization construction?

Assume $X$ is a smooth projective curve of genus $g\geq 3$ over $\mathbb{C}$ and let $M$ be the (singular) moduli space of semistable rank two vector bundles with trivial determinant on $X$. Then ...
6
votes
2answers
370 views

Weighted projective spaces as stacks

As stacks are the weighted projective line $\mathbb{P}$(1,n-1) and $\mathbb{P}$(k,n-k) isomorphic? Is there any reference for this?
3
votes
1answer
105 views

On factorization theorem of toric birational morphisms

Let $X_{Σ′}\to X_{Σ}$ be a toric birational morphism between smooth and complete toric varieties induced by a regular subdivision $Σ′\leq Σ$, i.e. every cone in $Σ′$ is contained in a cone in $Σ$ and ...
0
votes
2answers
158 views

Existence of $B$-reduction of a $G$-torsor on a curve

Let $k$ be an algebraically closed field, $X$ a connected smooth curve over $k$, $G$ a connected reductive group over $k$, and $B \subset G$ a Borel subgroup. Given a $G$-torsor $E$ on $X$ in the ...