Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,608
questions
2
votes
1
answer
180
views
Perfect complexes of plane nodal cubic curve
Let $C\subset\mathbb{P}^2$ be a plane nodal cubic curve with a unique singular point $O$ at the origin. Then I consider its normalization, denoted by $\widetilde{C}$ and let $\pi:\widetilde{C}\...
- 1,942
7
votes
2
answers
559
views
A curious equation on determinant----linear algebra or algebraic geometry?
I recently find a curious and unexplainable(as seems to me) equation on determinant as follows.
$$3\begin{vmatrix}
a_1 & b_1 & c_1 & d_1 \\
a_2 & b_2 & c_2 & d_2 \\
...
- 305
7
votes
1
answer
1k
views
Which revolutions in topology and geometry can we expect in the next 20 years? [closed]
In my limited perspective on the history of mathematics, I can name at least two big revolutions in Topology and Geometry (broadly construed): the introduction of Schemes in Algebraic Geometry, and ...
1
vote
0
answers
107
views
Nice, concrete example of pl-flipping contraction
In a course I'm giving on the MMP, I am discussing the importance of Shokurov's notion of a pl-flipping contraction for showing that flips exist for arbitrary flipping contractions. Does someone have ...
- 2,098
4
votes
0
answers
210
views
Classification of affine varieties over the affine line whose central fiber is $\mathbb{C}$ and general fiber is $\mathbb{C}^*$
Consider an affine variety $Y$ equipped with a morphism $\pi: Y \rightarrow \mathbb{C}$. The conditions we have are that $\pi^{-1}(0)=\mathbb{C}$, and for any $x$ not equal to zero, $\pi^{-1}(x)=\...
- 303
2
votes
0
answers
150
views
When is $D^+(QC(X))$ not the same as $D_{qc}(X)^+$ for schemes?
Let $QC(X)$ be the abelian category of quasicoherent sheaves on a scheme $X$. There is a functor
$$D^+(QC(X)) \to D_{qc}(X)^+$$
which is an isomorphism if $X$ is Noetherian or quasi-compact with ...
- 2,035
7
votes
1
answer
374
views
Relationship between Serre-Tate coordinates of ordinary elliptic curves and Tate curves
Let $K$ be a complete extension of $\mathbb{Q}_{p}$ with valuation $v$ over $p$, valuation ring $R$, maximal ideal $\mathfrak{m}$ and residue field $k$. It is well known that if $E/K$ is an elliptic ...
- 153
0
votes
1
answer
135
views
On the solutions of system of homogeneous polynomials of degree $d$ in $n$ variables
Consider the following two system of n homogeneous polynomials in n variables of degree $d$ with complex coefficients:
System 1 ($S_1$):
$f_1(x_1,\dots,x_n) = 0$,
$f_2(x_1,\dots,x_n) = 0$,
$\vdots$
$...
- 1,219
3
votes
0
answers
140
views
Locus where a family of cycles is rationally trivial is countable union of closed subvarieties?
Following up on this question which received a negative answer, I wonder if something weaker is true.
We work in the same set-up as the previous question. Let $B$ be a smooth quasi-projective variety ...
- 949
3
votes
0
answers
171
views
Does "derived" make anything constant in non-flat families?
This is an extremely basic (and surely amateurish) question that might be about derived geometry.
In usual algebraic geometry, if we have a flat projective morphism $f:X \to S$ with $S$ integral, and ...
- 91
3
votes
0
answers
73
views
Let $f: X \to D$ be a proper holomorphic submersion. Does a holomorphic form on $X$, closed on each fiber of $f$, have holomorphic coefficients?
Let $D$ be the unit disc in $\mathbb{C}^n$ and let $f: X \to D$ be a proper surjective holomorphic submersion, which is trivial as a smooth fiber bundle, with connected fiber $F$. We get an induced ...
- 487
1
vote
0
answers
94
views
Cokernel of map of dual of sheaves of differentials/deformations
Let $C$ be a nodal projective curve over an algebraically closed field of genus at least $2$. There are two natural "differential objects" one can consider: The sheaf of differentials $\...
- 203
0
votes
0
answers
116
views
Is any singularity a subgerm of $(\mathbb{C}^n, 0)$?
I am studying singularity theory. I have often come across, in the literature, the sentence which says "let $(X,0) \subset (\mathbb{C}^n,0)$ be a singularity". Here a singularity is a ...
- 369
3
votes
2
answers
145
views
Affine structure on the circle whose atlas consists of homeomorphisms onto $\mathbb{R}$
Let $S^1$ be the unit circle in the complex plane.
An atlas on $S^1$ is a finite collection $\alpha = \{ (U_i, \phi_i)\}_{i=1,\ldots,n}$ of pairs, where $U_i$ is an open subset of $S^1$ and $\phi:U_i\...
1
vote
0
answers
59
views
A criterion for divisors of degree $n$ on the projective line to belong to a linear system of codimension 1
My question is essentially about linear dependence/independence of polynomials, but I will formulate it in the language of algebraic geometry, hoping someone may suggest a result in algebraic geometry ...
- 5,011
0
votes
1
answer
203
views
Some questions about splitting of sequence $0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$ for Henselian val field $K$
I have a couple of questions about following proof by Peter Scholze on splitting of the ses (...does it have a name?...)
$$0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$$
for $K$ henselian valuation ...
- 6,000
2
votes
0
answers
201
views
Using the Dold-Thom Theorem to define \'etale cohomology
For reasonable spaces $X$, the Dold-Thom Theorem states that $\pi_i(SP(X)) \cong \tilde{H}_i(X)$ where $SP(X) = \bigsqcup_i \mathrm{Sym}^i(X)$. There is a purely algebro-geometric realization of this ...
- 7,646
2
votes
0
answers
67
views
Fundamental group of punctured strict henselization of a relative affine line over a non-regular base
Let $R$ be a local ring with maximal ideal $m$ of residue characteristic $p\ge 0$. Let $A$ be the strict henselization of $R[t]$ at the ideal $(m,t)$.
What is the maximal prime-to-$p$ quotient of the ...
- 7,161
1
vote
0
answers
55
views
Complex geodesic coordinate, local ramified map, and the conic metric
Remark: I have asked this question in MSE, however, I got no responses. This is the reason I come to ask here. I am looking forward to some advices. Thanks in advance
Let $X$ be a compact Kaehler ...
- 325
5
votes
0
answers
275
views
Formal neighborhood of stable curves
For a smooth projective curve $X/\mathbb{C}$, every (infinitesimal) deformation is trivial when restricted to $X \setminus x$ for any $x \in X$. In particular, all deformations can be obtained by “...
- 732
4
votes
1
answer
348
views
Sufficient condition for a probability distribution on $\mathbb Z_p$ to admit a square-root w.r.t convolution
Let $p \ge 2$ be a positive integer, and let $Q \in \mathcal P(\mathbb Z_p)$ be a probability distribution on $\mathbb Z_p$.
Question. What are necessary and sufficient conditions on $Q$ to ensure ...
- 6,726
3
votes
1
answer
232
views
Symmetric differential forms on moduli space of curves
Do there exist regular symmetric differential forms on $\overline{\mathcal{M}}_{g,n}$ the DM-stack of stable genus $g$ curves with $n$ marked points? By this, I mean nonzero sections $$ \omega \in H^0(...
- 3,301
5
votes
0
answers
226
views
Locally acyclic morphism which is not flat
Let $k$ be a closed field of characteristic $p \geqq 0$ and $\Lambda = \mathbf{Z}/\ell$, $\ell \neq p$. Recall that a morphism $f \colon X \to S$ of $k$-varieties is said to be locally acyclic if for ...
- 253
0
votes
0
answers
128
views
A stalk criterion for unit map to be an isomorphism on étale site
Let $f: X \to Y$ be a morphism of schemes and $\mathcal{F}$ sheaf of sets/Abelian groups on the small étale site $Y_{ét}$. Assume we manage somehow to show thatat every geometric point $\overline{y} \...
- 6,000
1
vote
1
answer
234
views
Restriction of reflexive sheaves
Let $X$ be a normal projective variety and let $\mathcal{L}$ be a rank 1 reflexive sheaf on $X$. Let $Y$ be a closed normal projective subvariety in $X$ such that the singularities of $X$ intersect $Y$...
- 1,750
7
votes
0
answers
181
views
Are the spaces BG for compact connected groups G ind-projective or ind-Kähler?
Let $G$ be a compact connected group, or maybe better its complexification. By thinking about the simplicial Borel space, or using $n$-acyclic $G$-spaces for higher and higher $n$, it's "easy&...
- 44k
1
vote
0
answers
331
views
Amitsur's theorem for generalized Severi–Brauer varieties
Let $k$ be a field of characteristic zero and assume that $A$ is a central simple algebra of index $2^n > 2$. We denote by $\operatorname{SB}_i(A)$ the $i$-th (generalized) Severi–Brauer variety of ...
- 1,419
2
votes
0
answers
96
views
Equivariant line bundles over toric variety
Let $X$ be a projective $n$-dimensional toric variety acted by an algebraic torus $T\simeq \mathbb{C}^{\ast n}$. It is well known that any piecewise linear (and integer in some sense) function on ...
- 21.1k
8
votes
1
answer
599
views
A question on algebraic independence
Let $f_1,f_2,\ldots,f_n, g \in \mathbb{F}_q[x_1,...,x_m]$. Assume that $f_1,\ldots,f_n$ vanish at $0$, so that $\mathbb{F}_q[[f_1,...,f_n]]$ is a subring of $\mathbb{F}_q[[x_1,...,x_n]]$. Suppose that ...
- 159
3
votes
1
answer
185
views
Čech cohomology refinement mapping
Let us consider the map $t_{AB}^*:H^1(A,F)\to H^1(B,F)$ between the cohomology groups, induced by the refinement map $t_{AB}:J\to I$, where $F$ is a sheaf of abelian groups on $X$, $A$ and $B$ are ...
0
votes
1
answer
88
views
Construct next polynomial from predecessor and resulting GCD
I have a sequence of polynomials built from an interpolation derived in a combinatorial problem. For each integer value of a parameter $n$ there is a different polynomial.
After trying to find a way ...
- 61
0
votes
0
answers
205
views
Definition of motivic/coming from geometry
Suppose $X$ is a projective smooth connected curve, $S\subset X$ is a finite set of points and $U=X\setminus S$.
I encountered the following definiton:
We say a $\mathbb Q$-local system $\mathcal F$ ...
- 523
0
votes
1
answer
116
views
Integer quadratic representation subject to discriminant minimization algorithm
Let $f(x)=ax^2+bx+c$ and $f(x)=n$. Is there an algorithm to choose $a,b,c$ such that the discriminant is minimized? Where $a,b,c,n,x$ are all integers.
More concretely, is there an algorithm to find $...
1
vote
1
answer
227
views
Zeroes of elementary polynomials without involving closed-form solutions
Consider the following two polynomials, where $n$ is an integer:
$$
p_n(x) = x^3-\frac1nx-\frac2n, \\
q_n(x) = x^2-\frac2n.
$$
For any $n$, let $x_p=x_p(n)$ and $x_q=x_q(n)$ be the unique positive ...
- 21
3
votes
1
answer
423
views
Characterization of étale locally constant sheaves over a normal scheme
I have a question about the verification of remark 1.2 in James Milne's book Étale Cohomology stated on page 156:
Assume $X$ be a normal & connected scheme with generic
point $g: \eta \to X$.
Then ...
- 6,000
1
vote
1
answer
128
views
sheaves on products and restrictions to fibres
Let $S$ be a smooth curve and $X$ a noetherian scheme (over $\mathbb{C}$). Let $\Sigma\subset S$ be a nonempty set of closed points. Let $\mathcal{E}$ be a coherent sheaf on $X\times S$. We have then ...
- 729
3
votes
1
answer
380
views
F-crystals from crystalline cohomology
In Section 7 of Katz' paper:
https://web.math.princeton.edu/~nmk/old/travdwork.pdf
He asserts "Crystalline cohomology tells us that for each integer $i \geq 0$, the de Rham cohomology $H^i=Rf_*(\...
- 459
2
votes
2
answers
270
views
Involution of the symmetric square of a smooth plane quartic
Let $C$ be a smooth plane quartic defined over a field $K$. Denote by $J$ its Jacobian, and by $C^{(2)}$ its symmetric square. Since $C$ is a smooth plane quartic, it is non-hyperelliptic, and hence ...
- 1,641
3
votes
0
answers
109
views
On the conditions for Artin-Schelter Gorenstein algebras
Let $ k $ be a field and $ A $ a connected graded $ k $-algebra ($ A $ is associative, but not assumed to be commutative).
The algebra $ A $ is called Artin-Schelter Gorenstein* of dimension $ d $ if ...
- 886
6
votes
0
answers
195
views
Ranks of elliptic curves over cubic fields
We are writing a paper on the ranks of elliptic curves over cubic fields. The curves of different torsion subgroups are created by the formulas in Jeon et al. and by our new parametrizations.
D. Jeon,...
- 913
0
votes
0
answers
59
views
What is the lattice point distribution over binary quadratic forms?
Let $f(x,y)=x^2+ny^2$ be the binary quadratic form of interest and consider the lattice points $S=\{ (x,y,f(x,y)) \in \mathbb{N}^3 \}$.
For simplicity, we keep things only on quadrant I of the ...
7
votes
1
answer
196
views
Explicit equations for the universal vector extension of an elliptic curve
The universal vector extension $E$ of an abelian variety $A$ is an algebraic group, an extension of $A$ by a vector group $0 \to V \to E \to A \to 0$, such that any other extension of $A$ by a vector ...
- 487
3
votes
0
answers
157
views
Action of complex Lie group on Dolbeault cohomology
Let $M$ be a compact complex manifold acted holomorphically by a complex Lie group $G$. Let $F$ be a holomorphic $G$-equivariant vector bundle over $M$.
Consider the natural representation of $G$ in (...
- 21.1k
2
votes
0
answers
233
views
Action of algebraic group in cohomology of equivariant algebraic vector bundle
Let $X$ be a projective algebraic variety over an algebraically closed field. Let an algebraic group $G$ act algebraically on $X$. Let $\mathcal{F}$ be a $G$-equivariant vector bundle (or, more ...
- 21.1k
4
votes
3
answers
290
views
How to recover integer part from known fractional root part?
Suppose you have $r=n+f$ where $n\in\mathbb{N}$ and $f\in (0,1)$. I know that $r^2$ is an integer and I can also get as many digits of $f$ as I like, is there a way to recover the value of $n$?
Thank ...
1
vote
0
answers
70
views
Increasing coverings of rigid analytic varieties
Let $K/\mathbb{Q}_p$ be complete and let $X/K$ be a rigid analytic variety. When does $X$ admit an "increasing" admissible covering by quasi-compact admissible (in the strong G-topology) ...
- 36
0
votes
0
answers
119
views
Pull-back of a divisor via composition with a rational map
Let $X,Y,Z$ be smooth projective varieties (say, over $\mathbb{C}$.) Let $f:X\to Y$ and $h:X\to Z$ be surjective morphisms and let $g:Y\dashrightarrow Z$ be a surjective rational map (not just ...
- 809
26
votes
4
answers
2k
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 ...
- 4,353
2
votes
0
answers
77
views
Projective fibrations between quasi-projective varieties
Let $f\colon X\to Y$ be a smooth surjective morphism between smooth quasi-projective varieties over $\mathbf{C}$. Assume that $f^{-1}(y)\cong \mathbf{P}^1$ for any closed points $y\in Y$.
Question: Is ...
- 505
1
vote
1
answer
182
views
Sufficient condition such that the set of zeros of an analytic function $f:\mathbb{R}^n \to \mathbb{R}$ contains only isolated points
Consider a real- analytic function $f: \mathbb{R}^n \to \mathbb{R}$. We know that zeros of $f$, roughly speaking, live in the low dimensional manifolds.
My question: Does a 'reasonable' sufficient ...
- 631