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

**8**

votes

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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 ...

**-5**

votes

**0**answers

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

**0**answers

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

**2**answers

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

**2**answers

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

**2**answers

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

**1**answer

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**

vote

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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**

vote

**0**answers

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**

votes

**0**answers

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**

vote

**0**answers

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

**0**answers

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**

votes

**0**answers

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

**1**answer

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 ...

**-4**

votes

**0**answers

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

**1**answer

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 ...

**-2**

votes

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**3**answers

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

**0**answers

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

**1**answer

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 ...