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_{...
