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

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8
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0answers
281 views

Meaningful review of Moriwaki's “Arakelov Geometry”

I have been asked to write a mathscinet review for Atsushi Moriwaki's Arakelov Geometry book: http://www.ams.org/bookstore-getitem/item=mmono-244 I could do the review the standard way in a day or ...
1
vote
1answer
146 views

A generalization of miracle flatness theorem

I wonder if the miracle flatness theorem Generalizing miracle flatness (Matsumura 23.1) via finite Tor-dimension still works if the rings involved are not local (and the dimension condition is ...
-1
votes
0answers
34 views

Coefficients of the pull-backs of divisors by resolving morphism

Let $\varphi : X \dashrightarrow X$ be a rational map. By a theorem of Hironaka we can find a resolution of singularities $(\tilde{X}_\varphi,\pi)$ of $\varphi$, where $\tilde{X}_\varphi$ is a ...
3
votes
1answer
488 views

Why we study Geometric invariant theory?

I am trying to learn Geometric invariant theory like it was introduced by Mumford. But I do not have a strong motivation and so I want to know the reason of studying Geometric invariant theory. I just ...
0
votes
0answers
139 views

Why the geometry of pair $(X,D)$ is important [on hold]

Let $(X,D)$ be a pair where $X$ is an algebraic variety and $D$ is a divisor on $X$. Why the geometry of pair $(X,D)$ is important?
2
votes
0answers
97 views

A multivariable polynomial degree question

Given $\mathsf{F,G}\in\Bbb R[x_1,\dots,x_n]$, minimum multivariate polynomials of least total degree $\mathsf{degF}$, $\mathsf{degG}$ such that, given unequal $a,b\in\Bbb R$, $$\mathsf{F(p)}=a, ...
1
vote
0answers
89 views

Characterizations of regular holonomic D-modules

I'm looking for references for the various characterizations of regular holonomic D-modules, in particular proofs of their equivalence. For instance, some characterizations I've seen (in the analytic ...
1
vote
1answer
137 views

Dimension of Ext modules [on hold]

Let $(R,m)$ be a noetherian local ring, and $M$ and $N$ be two finitely generated $R$-module. Then is it true that $\dim \text{Ext}^k(M,N)\leq \dim M-k$? If not does the reversed inequality hold?
2
votes
1answer
138 views

Second cohomology groups of Nakajima quiver varieties

Let $X=M(v,w)$ be a Nakajima quiver variety for a quiver $Q$. Can one calculate the second singular cohomology groups $H^2(X,\mathbb Z)$ or $H^2(X,\mathbb C)$ explicitly, and if not, are there some ...
1
vote
0answers
188 views

Algebraicity of the stack of coherent sheaves

I am trying to understand the proof of Theorem 4.6.2.1 in the book on algebraic stacks by Laumon and Moret-Bailly. The setting is this: $S$ is a Noetherian scheme, $f\colon X \rightarrow S$ is a ...
8
votes
0answers
148 views

Higher Fano varieties and Tsen's theorem

The rational connectivity of (complex) Fano manifolds ($c_1(T_X) > 0$) is one of the major, and surely most memorable achievements of Mori's bend-and-break method. To this day, despite intensive ...
1
vote
1answer
161 views

symplectic reduction for pair $(M,D)$

Let $M$ be a symplectic manifold with divisor $D$. Then how can we define symplectic reduction for pair $(M,D)$?
6
votes
0answers
210 views

What is an excellent algebraic space?

What does it mean to say that an algebraic space $S$ is excellent? One knows that excellence of a Noetherian ring is not a property that is etale local (that is, excellence cannot be checked over an ...
1
vote
1answer
81 views

Induced topology on site + Reconstructing global sections of a scheme (Orlov)

Let $(C,T,O)$ be a ringed site. Let $X$ be a presheaf on C. We get an induced ringed site $(C/X,T_X,O_X)$. C/X is the over category wrt the presheaf X. The topology $T_X$ is the biggest topology ...
2
votes
0answers
100 views

Do we have the following “devissage commutative diagram” in K-theory?

Let $X$ be a non-reduced Noetherian scheme. We define $K^0(X)$ to be the Grothendieck group of the derived category $Perf(X)$ and $K_0(X)$ to be the Grothendieck group of the derived category ...
9
votes
0answers
125 views

Why are unramified maps not required to be locally of finite presentation?

I have read and heard several times that it is “important” that unramified maps are not required to be locally of finite presentation, but only locally of finite type. Apart from this issue with ...
0
votes
0answers
74 views

The injection of direct image sheaf

Let $f:X \longrightarrow Y$ be a smooth holomorphic fibration between K\"ahler manifolds and $L$ be a holomorphic line bundle on $X$. Let $m$ be a positive integer. We denote by ...
1
vote
1answer
137 views

Could we extend any line bundle on the smooth part of a singular curve to a line bundle on the whole curve?

Let $X$ be a singular curve over an algebraic closed field $k$ with characteristic zero. Let $Z$ be the closed subset of singular points on $X$ and $U=X-Z$ be the smooth part, which is an open subset ...
0
votes
0answers
50 views

Is a general extension of general stable sheaves on $\mathbb P^2$ stable?

Theorem 2 in this paper by Bhosle gives a nice condition on slopes for when a general extension of general stable bundles on curves is stable. Does anyone know whether there is an analogous result for ...
5
votes
1answer
142 views

Existence and uniqueness of extensions of a finite flat map

Suppose that $S$ is smooth and that $U\subset S$ is a dense open subscheme. Let $X$ be a scheme (not necessarily smooth) and let $f:X\to U$ be a finite flat morphism. I would like to know whether ...
0
votes
1answer
71 views

A question of direct image of relative canonical bundle

Proposition: Let $f:X\rightarrow Y$ be a smooth holomorphic fibration between K\"ahler manifolds, and $L$ be a holomorphic line bundle. Then there exists a Zariski open set $Y_0\subset Y$ such that ...
9
votes
1answer
312 views

Can every algebraic variety of dimension $n$ be covered by $n+1$ affine opens?

Suppose $X$ is a complete algebraic variety of dimension $n$. Must there exist an affine covering with $n+1$ pieces? (For a projective variety in $\mathbf{P}^m$, we can always project it to some ...
-1
votes
0answers
53 views

Degree of function field extension in several variables (degree of an endomorphism over an AV)

I just want to know which is the best way to calculate the degree of a function field extension like this $[\mathbb{F}_q(a,b,c):\mathbb{F}_q(x,y,z)]$ where $x\mapsto f(a,b,c)$ $y\mapsto g(a,b,c)$ ...
3
votes
0answers
91 views

Action of automorphisms on cohomology with supports

Let $x$ be the closed point of an $n$-dimensional local scheme $X$, essentially smooth over a field $k$. Let $M$ be a sheaf on the category of smooth $k$-varieties (in either Zariski or Nisnevich ...
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votes
0answers
112 views

Must read books on topics in IMO [on hold]

I know there was this post Good books on problem solving / math olympiad However, I was looking for books that don't just approach in problem solving, but talk about the development of the theory. ...
0
votes
0answers
169 views

Complete Intersection

Let $I$ be an ideal of the polynomial ring $P=K[x_{1},...,x_{n}]$ that is generated by degree two polynomials ${f_1,...,f_k}$. The zero set $\mathcal{Z}(I)$ is isomorphic to an affine space of ...
3
votes
2answers
186 views

What are the easiest examples of irreducible, but not big, monodromy representations

Let $\rho: \pi_1(S,s_0) \to GL(V)$ be the monodromy representation associated to a local system of $\mathbb Q$-modules $\mathbb V$ with $\mathbb V_{s_0} = V$. Let $H$ be the Zariski closure of the ...
5
votes
2answers
259 views

References for the moduli space of complex structures

I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex ...
1
vote
2answers
160 views

Examples of quotients by infinitesimal group schemes

I'm looking for examples of explicit actions of the infinitesimal group schemes $\alpha_{p^n}$ on schemes (maybe as simple as the affine plane) in characteristic $p$ or mixed characteristic, and their ...
2
votes
1answer
155 views

Automorphisms of complete local rings

Let $k$ be a field and $(A,m)$ be the completion of the local ring of a smooth point of a $k$-variety. Let $x_1,x_2\in m\backslash m^2$ be regular elements. I am interested in knowing if one can find ...
1
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1answer
72 views

Multiplicity of the intersection of a Rational curve in a quadric with a tangent plane

Consider a rational map $u : \mathbb{CP}^1 \to \mathbb{CP}^4$ of degree~$d$, such that the image lies in a fixed 3-dimensional quadric $Q^3$. In other words, its image is a rational curve in $Q^3 ...
-4
votes
0answers
77 views

Showing two Rings are nor isomorphic [on hold]

I have the two rings $R[x,y]/(x^2+y^2-1)$ and $R[x,y]/(x^2-y^2-1)$ and I am trying to show they are not isomorphic over the real numbers. I have been playing around and I got that each polynomial in ...
25
votes
1answer
862 views

Combinatorics of K(Z,2)?

Anybody knows a semi-simplicial model for $K(Z,2)$ having finite number of simplexes in any dimension? With some regular description? I have heard about big activity on triangulating $CP^n$ but this ...
0
votes
0answers
149 views

Determine existence of irreducible variety in given homology class

Given a homology class $\alpha \in H_k(X,\mathbb{Z})$ on a variety $X$, is there a way to determine if there exists an irreducible subvariety $Y \subset X$ that has that class, i.e. $[Y] = \alpha$? ...
-1
votes
0answers
88 views

Is the cusp point of the curve $y^2=x^3$ a regular embedding? [closed]

This question maybe trivial. Let's consider the curve $y^2=x^3$ over a field $k$. It seems to me that the point $(0,0)$ is a regular embedding of codimension $1$ because it is given by the equation ...
1
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0answers
66 views

Is there any explicit result on the triangulated category of singularities of a curve?

This question is related to this MO question. Let $X$ be a projective curve over a field $\mathbb{C}$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category ...
0
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0answers
65 views

Pull-back of reflexive sheaves

Let $X$ be a noetherian, projective scheme, $\mathcal{F}$ be a reflexive sheaf on $X$ pure of dimension $\dim(X)$ and $Y \subset X$ be a closed subscheme of $X$. Is it possible that the pull-back of ...
1
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0answers
27 views

Lattice-isotopic essentialization of arrangements

I'm working on a problem related to $\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof ...
0
votes
0answers
61 views

Is there an explicit way to glue a stable map in projective space by writing down the family of maps explicitly in terms of polynomials?

Let $v_1:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ and $v_2:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be two holomorphic maps of degree $d_1$ and $d_2$ respectively. Suppose they agree at some ...
0
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0answers
42 views

compute standard basis in local rings

Let$>'$ be the order in $k[t,x_1,\cdots,x_n]$ as follows: Each semigroup order > on monomial in the $x_i$ extends to a semigroup order >' on monomial in $t,x_1,\cdots,x_n$ in the following way. We ...
1
vote
1answer
336 views

Atiyah's vector bundles over an elliptic curve

I'm reading through Atiyah's paper that classifies vector bundles over an elliptic curve, and I'm a little confused about one of his proofs. Lemma 15(i) states that if $E \in \mathcal{E}(r,d)$ is a ...
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votes
0answers
41 views

Caledonian college level2 [closed]

assume we have a stick of one meter length. we put 999 ants on the stick, at arbitrary positions and arbitrarily facing either left or right. At a certain time, all ants start moving with the same ...
4
votes
1answer
96 views

Singularities of the moduli stack of polarized hyperkahler varieties

Inspired by the recent question on singularities of the moduli stack of Calabi-Yau threefolds (Singularities of the moduli stack of Calabi-Yau threefolds) I'd like to ask the following question. Is ...
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0answers
66 views

Why are algebraic cycles rational? [migrated]

Let $X_{/\mathbb{C}}$ be a projective non-singular variety of dimension $n$ and $Z \subset X$ be an irreductible subvariety of dimension $p$. Denote by $\mathrm{H}_{\mathrm{dR}}^i(X,\mathbb{C})$ the ...
1
vote
0answers
54 views

Twisting stable maps to C* equivariant space by a line bundle

Let $X$ be a $\mathbb{C}^*$-equivariant algebraic variety. Then there is a notion of a map to $X$ twisted by a line bundle. Namely, let $B$ be a variety and $L/B$ a line bundle. Let $P_L=L\setminus ...
2
votes
1answer
229 views

The topology of Fano schemes of lines

Is there any references concerning the computation of the fundamental groups and Hodge numbers of Fano schemes of lines in a smooth hypersurface in $\mathbb{P}^n$?
0
votes
0answers
88 views

Is it possible to find an explicit definition of the “universal” (co)tangent bundle?

Let $H_{0,1}(\mathbb{P}^2, d)$ be the space of holomorphic degree $d$ maps (that are not multiply covered) from $\mathbb{P}^1$ to $\mathbb{P}^2$ with one marked point $y \in \mathbb{P^1} $ ...
2
votes
3answers
162 views

ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
2
votes
0answers
64 views

first chern class versus compactifying divisor in Ramanujam's surface

I have an elementary question about Ramanujam's surface. Ramanujam's surface is naturally the complement of a singular divisor $D$ in the one point blow up of $CP^2$, $\mathbb{F}_1$. One can resolve ...
2
votes
1answer
121 views

Could we extend the exact sequence $K^0(X)\to K_0(X)\to K_0(D_{sg}(X))\to 0$ to the left?

Let $X$ be a variety over a field $k$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category of perfect complex $Perf(X)$. It is clear that $Perf(X)$ is a ...