All Questions
22,548 questions
0
votes
1
answer
178
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Cohomology of Blow-ups and Minimal Models in Higher Dimensions
Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic zero. Consider a sequence of blow-ups:
$$X_n \xrightarrow{\pi_n} X_{n-1} \xrightarrow{\pi_{n-1}} \cdots \...
- 17
2
votes
1
answer
132
views
What is the polarization type of the push-forward of the Poincaré-bundle to the Jacobian of a curve?
$\DeclareMathOperator{\Jac}{Jac}\DeclareMathOperator{\Pic}{Pic}$Let $C$ be a smooth curve of genus $g > 0$, and consider the Picard torus $\Pic^d(C)$ of line bundles of degree $d$. Let $\mathcal P$ ...
- 1,286
2
votes
0
answers
96
views
Confusion about the Lefschetz standard conjectures for abelian varieties in the integral setting
Let $(A,\theta)$ be a principally polarized abelian variety of dimension $d$ over a number field $k$. By the hard Lefschetz theorem
$$H^2(A,\mathbb{Q}_{\ell}) \xrightarrow{\theta^{d-2}} H^{2d-2}(A,\...
- 679
1
vote
0
answers
124
views
Space of all orthogonal partially complex $2\times2\times2$ tensors
I am trying to understand the space of all orthogonal tensors, I asked a more general version of this question here but with no solution yet found. I want to look at the simplest case first, namely a $...
- 101
14
votes
1
answer
763
views
Is 36 a sum of 4 rational fourth powers?
Hasse principle is known to hold for homogeneous quadratic equations, but fail for some 3- and 4-variable cubics, such as $5x^3+4y^3+3z^3=0$ or $15x^3+10y^3+4z^3+3t^3=0$. These counterexamples are ...
- 7,182
3
votes
1
answer
131
views
On Weil's theorem that a rational group action becomes regular action after some birational modification
People attribute the following theorem to Weil:
Any variety $X$ equipped with a birational action of a connected algebraic group $G$ is equivariantly birationally isomorphic to a variety $Y$ equipped ...
- 3,472
2
votes
0
answers
237
views
Obscure action of derivations on group schemes (SGA 3 Exp III)
In what follows, I will refer to prop. 0.8 in SGA 3 Exp. III which can be found for example at the link (https://webusers.imj-prg.fr/~patrick.polo/SGA3/). I will quickly introduce the notation without ...
- 21
4
votes
1
answer
236
views
Ampleness verifiable over faithfully flat cover
Let $X$ be a Noetherian scheme over a field $k$ and $\mathcal{L}$ an invertible sheaf. Recall $\mathcal{L}$ is called ample iff for every coherent $\mathcal{M}$ there exist a $n_0(M)$ such that for ...
- 5,966
1
vote
1
answer
211
views
Characterize descents of geometric finite étale cover by means of homotopy exact sequence
Let $X/k$ be a geometrically connected $k$-variety (=separated of finite type, esp quasi-compact; the base field $k$ assumed to be separable, so $\overline{k}=k^{\text{sep}}$), $\overline{X} := X \...
- 5,966
1
vote
0
answers
216
views
Dimension under change of ground field
I apologize if this question is "too elementary" - I am not an algebraic geometer. Are the following statements true?
Let $k\subset K$ an extension of algebraically closed fields of ...
- 19
0
votes
0
answers
31
views
Zero sets of sums of multivariate polynomials defined recursively with mutually disjoint support
It is difficult in general to say anything about the zero sets of a sum of two or more multivariable polyomials. However, I am interested in two special cases:
I have a family of multivariate ...
- 357
6
votes
1
answer
139
views
Quiver variety, generically symplectic
Theorem 11.3.1 (iv) of "Noncommutative Geometry and Quiver algebras" by Crawley-Boevey, Etingof and Ginzburg claims that, for a dimension vector $\mathbf{d}\in\Sigma_0$, the quiver variety $...
- 985
29
votes
4
answers
3k
views
What is the status of the theory of motives?
It has been almost 60 years since Grothendieck conceived the conjectural theory of motives in order to grasp the common behavior of the most important (Weil) cohomology theories.
But what is the ...
- 4,595
1
vote
0
answers
68
views
Uniqueness of a canonical homography decomposition
Consider a multi-camera system with $n \geq 3$ calibrated cameras, each represented by a projection matrix $P_i \in \mathbb{R}^{3 \times 4}$ for $i=1, \dots, n$. We first want to detect and track ...
2
votes
0
answers
146
views
Two open contractible subset of $\mathbb{R}^3$ which are not homeomorphic but are polynomial preimage of open sets of $\mathbb{R}$
About 2 decades ago I heared from some one that there are infinitely many open contractible subsets of space mutually non homeomorphic to each other. I confess that I do not remember the ...
- 366
40
votes
1
answer
5k
views
Clausen–Scholze's Theorem 9.1 of Analytic.pdf, in view of light condensed sets, AKA is the Liquid Tensor Experiment easier now?
In the recent lecture series run jointly from IHÉS and Bonn, Clausen and Scholze have reworked—again—their foundations of geometry to focus attention on not arbitrary condensed sets and solid modules ...
- 35.5k
3
votes
2
answers
342
views
Reference Request: Beilinson-Bloch conjecture in terms of Beilinson regulator isomorphism
I'm looking for a reference that provides a concise statement of the Beilinson-Bloch conjecture, specifically formulated in terms of an isomorphism under the Beilinson regulator map.
More precisely, I'...
- 2,907
1
vote
0
answers
124
views
Section of étale morphism of algebraic spaces
I am sorry in advance if this question is too naive for specialists. I just realized that I need it when doin research and I haven't taken any serious course on algebraic spaces. Let $u \colon U \...
- 893
3
votes
1
answer
265
views
Base change in Chriss-Ginzburg
Below is a fragment of the book by Chriss and Ginzburg. Proposition 5.3.15(b) is stated in $K$-theory. My question is, does the same conclusion (and proof?) of proposition 5.3.15(b) (i.e. base change) ...
- 2,974
4
votes
1
answer
308
views
Connectedness of Milnor fiber
Let $Q$ be a homogeneous polynomial in $n$ variables. Then it defines a locally trivial fiber bundle projection $Q:{\mathbb C}^n- Q^{-1}(0)\to {\mathbb C}-\{0\}$ (called Milnor fibration). Under what ...
- 585
8
votes
0
answers
259
views
What is an example of Beilinson's theorem on $D^b\mathrm{Perv}$ failing for non-field coefficients?
In Beilinson's paper "On the derived category of perverse sheaves", he proves that the realization functor $D^b\mathrm{Perv}(X,R)\to D^b_c(X, R)$ is an equivalence when $R$ is a field. Here $...
- 495
2
votes
0
answers
279
views
Why is the weight monodromy hard in mixed characteristics?
I know very little about the conjecture, beyond Grothendieck's monodromy theorem perhaps (a dense open subgroup of inertia acting unipotently on pure motives). But I heard that it was completely ...
- 2,907
7
votes
1
answer
845
views
Algebraic K-theory and Witt groups
Let $S$ be a ring with involution (with 2 invertible). Suppose that the non connective algebraic K-theory $K(S)$ is 0 (i.e. $K_{n}(S)=0$, for all $n$).
Can we say something about the (higher) Witt ...
- 855
1
vote
0
answers
178
views
Prop 1.3 in "Birational geometry of algebraic varieties": specialization of rational curves
I have a couple of questions about some arguments in proof of Proposition 1.3 from Birational geometry of algebraic varieties by Kollár and Mori (p 8):
Proposition 1.3. [Abh56, Prop. 4] Let $X$ be ...
- 5,966
2
votes
1
answer
344
views
Why this genus one curve over $\mathbb{F}_5$ appear to violate Hasse-Weil bound?
Working over $\mathbb{F}_5$, the affine curve $x^4+2=y^2$ has no points.
The projective curve $x^4+2y^4=z^2y^2$ has only one point $(0 : 0 : 1)$.
Both curves appear to violate Hasse-Weil bound of $4....
- 25.4k
3
votes
1
answer
232
views
Non-degeneracy in hyperplane intersections of canonical curves
Let $C$ be a smooth projective non-hyperelliptic curve over $\mathbb{C}$ of genus $g = 4$. The canonical bundle $\omega_C$ induces a canonical embedding $C \longrightarrow \mathbb{CP}^3 $ such that $C$...
- 343
1
vote
0
answers
143
views
Existence and injectivity of a map from $\mathrm{Num}(X) \otimes_\mathbb{Z} \mathbb{C}$ to $H^1(X,\Omega^1_X)$
I am currently reading a paper by Mustata and Popa on the Van de Ven theorem, you can read the paper here.
On page 51 there is the following map
$$\alpha_\mathbb{C} : \mathrm{Num}(X) \otimes_\mathbb{Z}...
- 201
0
votes
0
answers
98
views
Differential of the evaluation map of the Kontsevich moduli space
Let $X$ be a smooth projective variety, and $\beta$ a curve class on it. We have the Kontsevich moduli space $\overline{\mathcal M}_g(X, \beta)$ of stable maps from genus $g$ curves to $X$ with class $...
- 19
3
votes
1
answer
483
views
Can you build $\text{Aut}(X)$ using only $\text{QCoh}(X)$?
By the Gabriel-Rosenberg-Brandenberg-Gabber Theorem, sufficiently nice schemes $X$ are determined up to isomorphism by the category $\text{QCoh}(X)$.
My question: for sufficiently nice $X$, can you ...
- 5,711
252
votes
37
answers
179k
views
Best algebraic geometry textbook? (other than Hartshorne)
I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best.
Then what might be the 2nd best?
It can be a book, preprint, online lecture note, webpage, etc.
One suggestion ...
2
votes
1
answer
181
views
Idempotent algebras over absolutely flat ring
Is it possible to classify all idempotent algebras over an absolutely flat ring? Are there any idempotent $E_{\infty}$ algebras which are not discrete?
I am particularly interested in the special case ...
- 2,356
263
votes
29
answers
89k
views
Mathematical games interesting to both you and a 5+-year-old child
Background: My daughter is 6 years old now, once I wanted to think on some math (about some Young diagrams), but she wanted to play with me...
How to make both of us to do what they want ? I guess ...
2
votes
0
answers
126
views
Relative Jacobian as a ramified holomorphic quotient
Let $f:X \to S$ be an elliptic fibration with only $m$ singular fibers of type $I_1$ at the set of points $\lbrace s_1,\cdots, s_m \rbrace$ of $S$. In the paper "On Compact Analytic Surfaces: II&...
- 41
4
votes
1
answer
215
views
Proper morphism
Maybe this could be a silly question, but I am considering the following problem.
Let $G$ be a connected reductive group and $X$ be an affine $G$-variety, i.e., $G$ acts on $X$. (You can assume the ...
- 147
1
vote
0
answers
80
views
Computing Chow groups of affine, simplicial toric varieties
Let $k$ be an algebraically closed field. Let $X$ be an $n$-dimensional affine, simplicial toric variety over $k$. There exists an $n$-dimensional simplicial cone $\sigma$ in $\mathbb{R}^n$ such that $...
- 639
4
votes
0
answers
178
views
Splitting of counit-trace map for $\ell$-adic sheaf $\Bbb Q_{\ell}$
I have a question about following argument on page 3 in paper arxiv.org/abs/1702.04404 by Will Sawin (Proposition 3):
The claim is that for a finite, dominant map $f:X \to Y$ between varieties $X,Y$ ...
- 5,966
1
vote
0
answers
138
views
Quotients of open subsets of the semi-stable locus
This is a rewrite of a deleted question. I've decided to focus on one particular example mentioned in that question. Below a point means a closed point.
Let $U$ be the set of irreducible non-cuspidal ...
- 23.5k
2
votes
1
answer
204
views
What does the Serre functor of equivariant category of fractional CY category look like?
I am considering the following set up. Let $\mathcal{A}$ be a fractional Calabi-Yau category and denote by $S$ the Serre functor and $S^m=[n]$. Now I consider a finite group action $G$ on $\mathcal{A}$...
- 1,982
3
votes
0
answers
220
views
Computing pushforwards and pullbacks of D-modules
Let $X$ be a smooth algebraic variety (over some field of char 0), $Z$ a smooth closed subvariety of codimension 1, $i : Z \hookrightarrow X$ the inclusion, and $j : U \hookrightarrow X$ the ...
- 37k
17
votes
1
answer
783
views
Injective ring homomorphism from $\mathbb{Z}_p[[x,y]]$ to $\mathbb{Z}_p[[x]]$
Is there an injective $\mathbb{Z}_p$-ring homomorphism from $\mathbb{Z}_p[[x,y]]$ to $\mathbb{Z}_p[[x]]$?
- 2,907
4
votes
3
answers
266
views
References for $K$-orbits in $G/B$
Let $G$ be a reductive group, $K$ a symmetric subgroup of $G$ (e.g., fixed point of an involution), and $B$ a Borel subgroup of $G$. Then it is well known that $G/B$ has finitely many $K$-orbits. ...
- 741
2
votes
0
answers
157
views
Symmetric powers for a short exact sequence of vector bundles
If $0 \to A \to B \to C \to 0$ is a short exact sequence of vector bundles on $X$, is it necessarily true that $[{\rm Sym}^k B] = \sum_{i=0}^k [{\rm Sym}^i A \otimes {\rm Sym}^{k-i} C]$ in the ...
- 2,974
3
votes
1
answer
263
views
Could efficient solutions of $x^2+n y^2=A$ be related to integer factorization?
Let $n$ be positive integer with unknown factorization and $A$ integer with known
factorization.
According to pari/gp developers pari can efficiently find all solutions of:
$$x^2+n y^2=A \qquad (1)$$
...
- 25.4k
2
votes
0
answers
120
views
Looking at versions of Implicit Function Theorem (IFT) on rings
$ \let \ovr \overline
\def \Z {\mathbb Z}
\def \C {\mathbb C}
\def \F {\mathbb F}
\def \P {\mathcal P}
\def \x {\boldsymbol x}
\def \a {\boldsymbol a} $
Let $ \P = \{ p _ i ( \x , y ) \} _ { i = 1 } ^ ...
- 346
7
votes
3
answers
1k
views
Non-noetherian schemes with noetherian underlying space (in the Zariski topology)
I am curious if there are many non-Noetherian schemes with Noetheiran underlying space.
I know an example, we consider: $R= k[x_1, x_2, \dots]/\langle x_1^2, x_2^2, \dots\rangle $ for $k$ a field. ...
- 73
4
votes
1
answer
178
views
Computing the divisor class group of toric varieties over an arbitrary field
Let $k$ be an arbitrary field and let $X$ be a toric variety over $k$, coming from a fan $\Sigma$. If $k$ is algebraically closed, then theorem 4.1.3 of Cox ,Little and Schenck’s Toric Varieties book ...
- 639
1
vote
0
answers
42
views
Example polynomial system where Macaulay bound is tight
I have been solving systems of polynomial equations by forming the Macaulay matrix of different degrees and computing its null space. If the degree is large enough, namely at or above the degree of ...
- 11
2
votes
1
answer
68
views
When are solutions to a set of multi-variate quadratic equations isolated points?
Suppose I have set of $n$ multi-variate polynomial equations with $n$ unknowns $x_1, \dots, x_n$. The $n$ equations have real coefficients and are quadratic (so largest degree is $2$). How do I ...
2
votes
0
answers
58
views
$L^2$ approximation of delta functions on real algebraic varieties and asymptotic bounds
Let $X$ be a smooth projective variety over $\mathbb{C}$ of dimension $n$. Consider a probability measure $\mu$ on $X(\mathbb{R})$, absolutely continuous with respect to the Lebesgue measure induced ...
0
votes
0
answers
88
views
Geometry of prym locus
The celebrated solution to the Schottky problem provides a beautiful geometric characterization of Jacobians among all principally polarized abelian varieties (ppavs). One might hope for a similarly ...