for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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40 views

### Squarefree monomial ideals, locally complete intersections

Let $I\subset S=\mathbb{C}[x_0,\dots,x_n]$ be an ideal generated by squarefree monomials of the same degree. Is it true that $S/I$ is
a locally complete intersection?

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101 views

### Convergent sequences in projective varieties

It's very well known that if $X$ is an irreducible projective variety (feel free to assume that the base-field is $\mathbb{C}$), then any two points $x,y\in X$ can be connected by the image of a non-...

**7**

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176 views

### What is the mirror of an algebraic group?

Background: Kontsevich's homological mirror symmetry conjecture posits the existence of pairs $(X,\check X)$ with an equivalence of dg/$A_\infty$-categories
$$\mathcal F(X)=\mathcal D^b(\check X)$$
...

**1**

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**1**answer

141 views

### Some questions about Clemens' paper Cohomology and Obstructions I: Geometry of formal Kuranishi theory

I am reading the paper Cohomology and Obstructions I: Geometry of formal Kuranishi theory written by Herbert Clemens. I have found this paper hard to follow even though I am familiar with the ...

**1**

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78 views

### Holomorphic map, Instantons of Complex Projective Space and Loop Group

It seems that holomorphic (or rational) maps play a crucial role to relate the following data:
Instanton in 1-dimensional complex Projective Space $$\mathbb{P}^1$$
in a 2 dimensional (2d) spacetime.
...

**4**

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91 views

### The geometry and arithmetic of the intersection of a cubic and quadric threefold

Let $f,g \in \mathbb{Z}[x_0, \cdots, x_4]$ be a quadratic and cubic form (i.e., homogeneous polynomials) respectively, and let $V(f), V(g)$ denote their respective projective varieties; $V(f), V(g)$ ...

**3**

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**1**answer

141 views

### A possible “finite set avoiding” version of Grothendieck's Zariski Main Theorem?

Recall that in EGA $IV_3$ (Théorème (8.12.6)), Grothendieck calls the following theorem ``Zariski's Main Theorem":
Let $Y$ be a quasi-compact separated scheme, and $f : X \to Y$ is a
separated ...

**5**

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113 views

### Can free rational curves lift to ramified covers of Fano varieties?

Does there exist $X$ a smooth Fano manifold, $f: Y \to X$ a nontrivial ramified finite cover, $C \subseteq X$ a smooth very free rational curve, such that $f$ is étale over a neighborhood of $C$?
...

**13**

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**1**answer

682 views

### Does every sheaf embed into a quasicoherent sheaf?

Question. Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of $\mathcal{O}_X$-modules. Is there always a quasicoherent sheaf $\mathcal{E}'$ together with a monomorphism $\mathcal{E} \to \mathcal{E}'$?...

**5**

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137 views

### Infinite-dimensional manifold $ΩG$ of loops and the complex projective $n$-space

In a review seminar today, I heard the speaker takes the below for granted, but I have no enough background
"Yang-Mills (YM) instantons in 4D can be naturally identified with (i.e. have the same ...

**5**

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**2**answers

131 views

### Degree of the variety of independent matrices of rank $\leq r$?

Consider an $m$-by-$n$ matrix $A$ with entries in a field $k$; we can see $A$ as a point in the affine space $\mathbb{A}^{m n}$. The rank of $A$ will be $\leq r$ (where $r<\min(m,n)$) if and only ...

**3**

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77 views

### Periods for Irreducible Holomorphic Symplectic Manifolds

Let $f:\mathscr{X}\rightarrow \operatorname{Def(X)}$ be the Kuranishi family of $X$, where $X$ is an irreducible holomorphic symplectic manifold. After shrinking $\operatorname{Def}(X)$, we get that ...

**0**

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37 views

### Geometric intuition of the dimension of Grassmannians and flag manfolds [migrated]

I wish to understand geometrically (not just algebraically) why the dimension of the Grassmanian $G(k,n)$ is $k(n-k)$ and the dimension of a flag manifold $F(k_{1},k_{2},...,k_{n},N)$ is $\sum_{i=1}^{...

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83 views

### singularity of an irreducible surface in $\mathbb{P}^3$

Is it true that the singular locus of an irreducible hypersurface in $\mathbb{P}^3$ have pure co-dimension ?

**2**

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**1**answer

153 views

### Computation of cohomology of ideal sheaves

Let $j: X \to Y$ be a closed embedding. Let $I_{X/Y}$ be the ideal sheaf of this closed embedding. Then there is a exact sequence
$$ I_{X/Y} \to \mathcal{O}_Y \to j_{*}\mathcal{O}_X \to 0$$
One use ...