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

8
votes
0answers
30 views

The set of polytopes with given $f$-vector

Let $f=(f_0,\ldots f_n)$ be a vector in $\Bbb N^{n+1}$. Let $X$ be the set of all (ordered) $f_0$-tuples in $\Bbb R^n$ whose convex hull has $f$ as its $f$-vector. Assume that $X$ is non-empty. Is ...
3
votes
1answer
65 views

Derived functor giving algebraic map between moduli

Let $X$ be a smooth variety and $M$ a fine moduli space of certain kind of sheaves on $X$. Let $\mathcal{E}$ be the universal family on $X\times M$. Suppose there is a derived functor $F$ from $D^b(X)$...
5
votes
0answers
214 views

How to explain to an engineer what algebraic geometry is?

This question is similar to this one in that I'm asking about how to introduce a mathematical research topic or activity to a non-mathematician: in this case algebraic geometry, intended as the most ...
1
vote
1answer
68 views

Blow up of a Projective scheme along a projective-subscheme using Macaulay2

Given a closed-subscheme (corresponding to the homogeneous ideal $I$ inside $K[x_0,...,x_n]$) of a projective scheme $\mathbb{P}^n$, how can one find out the blow up using Macaulay2? ...
3
votes
0answers
123 views

holomorphic continuation of motivic $L$-functions

The question is rather easy to formulate: when is the $L$-function of a pure motive over $\mathbb{Q}$ expected to have a holomorphic (as opposed to simply meromorphic) continuation to the complex ...
2
votes
0answers
98 views

Does Leray Spectral sequence degenerates at $E_2$ over product of curves

Let $C$ be a smooth, projective curve (can assume to be rational) and $X:=C \times C$. Denote by $p:X \to C$ one of the two natural projections. Let $E$ be a vector bundle on $X$. Is it true that, $$...
2
votes
0answers
70 views

For a curve $C$ and its Jacobian $J$, is $\Gamma(J, \Omega) \to \Gamma(C, \Omega)$ an isomorphism?

Let $k$ be an algebraically closed field, $C$ a smooth complete $k$- curve, $K = k(C)$, $J$ its Jacobian variety, and $f : C \to J$ the morphism induced by some rational point of $C$. Then does $f$ ...
2
votes
0answers
80 views

Rational points on a weighted projective surface

The equation $$\displaystyle y^2 = f(x_1, x_2, x_3)$$ with $f$ a non-singular quartic form in three variables defines a del Pezzo surface of degree 2. I am interested in the similar construction $$\...
6
votes
0answers
90 views

Are dualizable objects in the derived category of a ringed topos perfect?

Recall that an object $a$ in a symmetric monoidal category $(\mathcal{C}, \otimes, e)$ is dualizable if there exists an object $b$ and morphisms $\varepsilon\colon b \otimes a \to e$ and $\eta\colon e ...
4
votes
2answers
201 views

How does multiplication affect degrees?

Let $M(n) \sim \mathbb{A}^{n^2}$ be the space of $n$-by-$n$ matrices, seen as an affine space over a field $K$, and endowed with the usual matrix multiplication. Let $V$ and $W$ be subvarieties of $M(...
2
votes
0answers
55 views

Compute the closure of graph of function from complement of hypersurface in $\mathbb{A}^n$

I'm hoping someone can give me some tips to help speed up computation on the following problem: Suppose I have a map $G=(g_1/f,\dots,g_m/f):\mathbb{A}^n\setminus{V(f)}\to \mathbb{A}^m$. I'm ...
2
votes
0answers
126 views

Complex projective algebraic variety, moduli space of flat connections, and instantons

In Looijenga's work below, if I understand correctly, it shows that Statement 1: At an algebraic variety, the moduli space of SU($N$) flat connections on a 2-torus $T^2$ is given by the space of ...
5
votes
0answers
136 views

Diffeomorphism type of Ricci-flat four manifolds

Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows: A) Is there a classification of the possible homeomorphism types of ...
5
votes
1answer
308 views

$BG$ the stack, $BG$ the simplicial presheaf

I have a theoretical question about comparing two objects that I have recently come across. For concreteness, let us work over the category $C$ of schemes over $k$. Let $G$ be an algebraic group over ...
3
votes
1answer
134 views

Known techniques to compute flat cohomology after base change

Let $f$ be some homogenous polynomial of degree $d>2$. Let $X = \operatorname{Proj} (k[x,y,z]/(f))$ where $k$ is algebraically closed field of characteristic $p >0$. Now let $R$ be a $k$-...
1
vote
0answers
69 views

Segre embedding and Hilbert polynomial of coherent sheaves

Let $X \subset \mathbb{P}^n$ and $Y \subset \mathbb{P}^m$ be smooth, projective subvarieties, $F$ and $G$ coherent, torsion-free, sheaves on $X$ and $Y$ with Hilbert polynomials $P_{F}$ and $P_G$, ...
3
votes
0answers
127 views

Fraction fields of strict henselizations of DVRs

Let $A_{1},A_{2}$ be discrete valuation rings whose fraction fields are isomorphic. Let $A_{i}^{\mathrm{sh}}$ be the strict henselization of $A_{i}$, and let $K_{i}$ be the fraction field of $A_{i}^{\...
6
votes
1answer
229 views

Cokernel of map of étale sheaves

Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ ...
0
votes
0answers
134 views

How to construct $ \mathcal{H} ( \mathbb{P}^n ) = \{ \ \text{Smooth projective subvarieties of } \mathbb{P}^n \ \} $?

Set : $$ \mathcal{H} ( \mathbb{P}^n ) = \{ \ \text{Smooth projective subvarieties of } \mathbb{P}^n \ \} $$ I would like to know if there exists a projective variety $ H ( \mathbb{P}^n ) $ whose ...
3
votes
1answer
97 views

The ample cone of a surface with an algebraic $\mathbb C^*$ action

Let $X$ be a compact complex protective surface that admits a nontirvial algebraic $\mathbb C^*$-action. It seems to me, that the ample cone of $X$ is polyhedral with finite number of faces. I wonder ...
6
votes
1answer
231 views

Cohomology of neighborhood of $\mathbb{C}\mathbb{P}^1$ in $\mathbb{C}\mathbb{P}^n$

Let $\mathbb{C}\mathbb{P}^1$ be embedded linearly to $\mathbb{C}\mathbb{P}^n$ with $n>1$. (Such an embedding is given in coordinates by $[x:y]\mapsto [x:y:0:\dots: 0]$.) Is it true that for any ...
3
votes
0answers
113 views

Cohomology of a tubular neiborhood of submanifold vs cohomology of the formal neighborhood

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $\hat Z$ be the formal neighborhood of $Z$. Let $U$ be an open neighborhood of $Z$. Is ...
2
votes
0answers
155 views

Torsion free-ness of cohomology of moduli of vector bundles

My question requires a little introduction: $\textbf{Atiyah-Bott's solution:}$ In the paper "The Yang mills equation of Riemann surfaces" Atiyah-Bott has computed the cohomology of moduli vector ...
6
votes
1answer
329 views

Classification of smooth algebraic surfaces with a smooth morphism to $\Bbb P^1$

Let $k$ be an algebraically closed field, it is well known that $\mathbb P^1$ is simply connected, but how about smooth projective surfaces $X$ with a smooth morphism to $\Bbb P^1$? Except the case $...
5
votes
1answer
404 views

Are there enough meromorphic functions on a compact analytic manifold?

Let $X$ be a compact complex analytic manifold, $D\subset X$ an irreducible smooth divisor, given as zeroes of a global meromorphic function $f\in {\mathfrak M} (X)$. Are there enough other ...
8
votes
0answers
311 views

The Grassmannian Gr(2,8) and an E7 surprise

Are there any mathematical explanations for the following surprising facts? $$\int_{Gr(2,8)} c_{\text{top}}(TX(-2)) = 6556 = \frac{1}{2} \deg(E_7/P(\alpha_7)) + 1,$$ and $$\int_{Gr(2,6)} c_{\text{top}}...
12
votes
1answer
483 views

Reference for the algebro-geometric proof of Matsumoto theorem

Matsumoto proved in his PhD thesis that if $F$ is a field then $$K_2(F)=(F^*\otimes F^*)/(x\otimes (1-x)).$$ The original Matsumoto proof as it is written in Milnor's book on algebraic K-theory looks ...
0
votes
0answers
73 views

Examples of degree zero, rank one reflexive sheaves without r-th roots

Let $X$ be a normal, projective surface (or more generally a variety) over $\mathbb{C}$ (i.e., $X$ is irreducible). Fix a polarisation on $X$. I am looking for examples of rank one, degree zero (...
3
votes
0answers
55 views

On ideals in Noetherian rings, isomorphic to the trace of some finitely generated module

Let $R$ be a Noetherian ring. For a finitely generated $R$-module $M$, let $tr_R(M):=Im(\tau_M)$, where $\tau_M:M\otimes Hom(M,R)\to R$ is the map defined as $\tau_M(m\otimes f)=f(m)$. Let $I$ be a ...
1
vote
1answer
180 views

fiber of a map into Grassmanian

Suppose $R\subset K=K_0\supset K_1\supset K_2\supset...\supset K_{n-1}\supset K_n=\{0\}$ are all vector spaces with $\dim R\cap K_i=r_i$ where $r_i$ are some fixed numbers. Suppose $O\subset Gr(r_0,\...
2
votes
0answers
87 views

Pontryagin square on spin and non-spin manifold

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ ...
8
votes
0answers
222 views

Cohomology of complex manifold vs cohomology of its complex submanifold

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that $$H^i(Z, A|_Z)=0 \mbox{ for any } ...
1
vote
2answers
131 views

Flatness of direct image sheaf over local artinian ring

Let $\pi:X \to \mbox{Spec}(\mathbb{C}[t]/(t^2))$ be a smooth, projective morphism and $L$ be an invertible sheaf on $X$. Denote by $L_0$ the restriction of $L$ to the closed fiber, say $X_0$ of $\pi$. ...
3
votes
1answer
201 views

Does a projective variety have only finitely many associated Hilbert polynomials?

Let $X$ be a projective variety over $\mathbb{C}$. If $L$ is an ample line bundle, then $h_L$ denotes the Hilbert polynomial. Is it true that, if $L$ and $L'$ are ample line bundles which are ...
22
votes
1answer
561 views

Is there an explanation of analogies between the cross-ratio and the Riemann curvature tensor?

Define the cross-ratio of four real or complex numbers as follows: $$[a,b,c,d] = \frac{(a-c)(b-d)}{(a-d)(b-c)}.$$ Then its logarithm has the same symmetries as the curvature tensor: $$\log[a,b,c,d] = -...
4
votes
1answer
173 views

How to check if a ternary cubic is a product of linear forms?

Let $F$ be a square-free (as a polynomial) ternary cubic form over $\mathbb{C}$, and let $H_F$ be its Hessian determinant... which is also a ternary cbic form. If $F$ splits over $\mathbb{C}$, so that ...
20
votes
2answers
961 views

Hodge theory (after Deligne)

In an interview with Deligne on the Simons Foundation website, I heard Robert MacPherson say that at the time Deligne's papers on Hodge theory were being published, the results seemed absolutely ...
4
votes
0answers
240 views

Bézout and products in algebraic groups

Let $G$ be a linear algebraic group -- be it a Lie group or a group of Lie type. Let $V$, $W$ be subvarieties of $G$. Of course, $V\cap W$ is also a variety (not necessarily irreducible) and $V\cdot W^...
2
votes
0answers
137 views

Is a reductive group scheme always parahoric?

Let $R$ be complete (or, more generally, Henselian) discrete valuation ring with fraction field $K$. Let $G$ be a reductive $R$-group scheme. Is $G$ a parahoric in the sense of Bruhat-Tits? If so, ...
0
votes
1answer
176 views

Naive question in Cech cohomology

Let $X$ be a smooth, projective variety and $F$ a coherent sheaf on $X$. Let $\{U_i\}_{i \in I}$ be an open affine covering of $X$ and $\{f_{ij}\}_{i<j}$ with $f_{ij} \in \Gamma(U_{ij},F)$ ...
3
votes
0answers
73 views

Semicontinuity of cohomology of torsion-free sheaves restricted to divisors

Let $X$ be a smooth projective variety, $\mathcal{E}$ a torsion-free coherent sheaf on $X$ and $\mathfrak{d}$ a linear system of divisors in $X$. I would like to show (at least when $X$ is a surface) ...
6
votes
1answer
184 views

To prove the fiber above a codimension 1 point contains a geometrically integral open subscheme

Suppose $f:X\rightarrow \mathbb{P}_k^n$ is a proper smooth morphism, where $k$ is an algebraically closed field. If $f$ admits a rational section, can we prove that the fiber of $f$ above any ...
1
vote
0answers
59 views

Explicit description of the scheme obtained by relative gluing data over a base scheme

I have recently been trying to get a better understanding of the projective space bundle of a quasi-coherent sheaf of graded algebras over a scheme $X$. The key idea is the following construction ...
5
votes
0answers
95 views

Explicit algebraic constructions of Parshin covers

Let $X$ be an algebraic curve defined over a number field $K$ and has genus $g \geq 2$. Let $P$ be a $K$-point of $X$. We say that $X_P$ is a Parshin cover of $X$ if $X_P$ is defined over a finite ...
4
votes
1answer
304 views

Kahler manifolds and algebraic varieties

Let $X$ be a smooth complete algebraic variety over $\mathbb{C}$. Can it happen that the underlying complex manifold is not Kahler? If yes, are there explicit examples? If not - how to prove this?
1
vote
0answers
95 views

Why is this the dualising sheaf of a singular curve?

If $X$ is a curve with a nodal singularity at $x$, it's referred to here and here that its dualising sheaf is $$\omega_X \ = \ \pi_*(\Omega_{X}(p_1+\cdots+p_n)').$$ Here, $\pi:X\to X'$ is the ...
7
votes
0answers
133 views

Maps between moduli of curves

Let $M_{g,n}$ be the moduli space of $n$-pointed curves, and $M_g[m]$ the moduli space of (unpointed) curves with $m$-level structure. Fix $m>0$. Is it true that for $n$ large enough, there is a ...
6
votes
1answer
189 views

How to visualize local complete intersection morphisms?

As the question title asks for, how do others visualize local complete intersection morphisms? My experiment in asking people in real life didn't pan out, so I'm consulting the MO algebraic geometry ...
1
vote
0answers
84 views

birational equivalence for log Calabi-Yau varieties

What I call a log Calabi-Yau variety is a smooth quasi-projective variety $U$ that admits a smooth compactification $X$ by a normal crossing divisor $D$ such that $-D$ is anticanonical. Or probably it'...
0
votes
0answers
55 views

A relation between an radical ideal and its $J$-radical

Let $R$ be a commutative ring with $1$. For an ideal $I$ of $R$ the $J$-radical of $I$, denoted by $J-rad(I)$, is the intersection of all maximal ideals of $R$ containing $I$, that is, $J-rad(I)=\cap_{...