All Questions
2,036 questions
3
votes
1
answer
493
views
What are meromorphic line bundles?
Initially I wanted to call this question "Categorification of meromorphic functions?" but discovered so many questions about categorification that I became scared and decided to replace it ...
- 18.4k
3
votes
0
answers
166
views
Cycle maps as edge maps
Given a smooth projective algebraic variety over $\mathcal{C}$, let $X$ be its associated complex analytic space.
The exponential sequence on $X$:
$$0\to\mathbf{Z}(1)\to\mathcal{O}_X\to\mathcal{O}_X^...
user119470
3
votes
1
answer
646
views
looking for an identity for higher jet bundle $J^kM$?
We know this fact that the first jet Bundle $J^1M$ is diffeomorphic with $T^*M×\mathbb{R}$.i.e,
($J^1M=T^*M×\mathbb{R}$)
Is there something like this identity for higher jet bundle $J^kM$?
I editted ...
user21574
3
votes
1
answer
423
views
Is it written anywhere that open subvarieties of affine spaces have "completely impure" cohomology?
Consider complex affine space $\mathbb{C}^n$ and let $U$ be a Zariski open subset of $\mathbb{C}^n$. By a celebrated result of Deligne, the cohomology $H^i(U)$ has a canonical Hodge structure. In ...
- 44.7k
3
votes
0
answers
375
views
Linear system on singular plane curve
Let $C \subset \mathbb{P}^2_k$ an irreducible plane curve of degree $d >1$
over algebraically closed field $k$. That is $C=V(f(x,y,z))$ where $f \in k[x,y,z]$ homogeneous of degree $d$. Let $\{...
- 5,946
3
votes
1
answer
438
views
Virtual Lefschetz motive
Hi there,
I have a question which popped up while reading papers on motives.
Let $V_k$ be the category of (projective) k-varieties, and let $K_0(V_k)$ be the Grothendieck ring of $V_k$; then $\...
- 4,585
3
votes
1
answer
364
views
General position argument for reasonable spaces
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of non zero homogeneous degree $d$ polynomials in three variables up to scaling, where
$\delta_d:= \frac{d(d+3)}{2} $. Given a point $p \...
- 3,245
3
votes
1
answer
2k
views
Inverse Function Theorem in Algebraic Geometry
Suppose that $X$ and $Y$ are smooth complex algebraic varieties, and that $f:X\rightarrow Y$ is an etale morphism in the sense that $d_xf:T_xX\rightarrow T_{f(x)}Y$ is an isomorphism for all $x\in X$. ...
- 4,920
3
votes
1
answer
2k
views
Why is this the dualising sheaf of a singular curve?
If $X$ is a curve with a nodal singularity at $x$, it's referred to here and here that its dualising sheaf is
$$\omega_X \ = \ \pi_*(\Omega_{X}(p_1+\cdots+p_n)').$$
Here, $\pi:X\to X'$ is the ...
- 5,711
3
votes
2
answers
512
views
Kernel of an integrable holomorphic dee-bar connection is a holomorphic vector bundle
I'm looking for references for the following fact, to pass on to a grad student. I think I can prove it, but it is a bit sloppy and I'd rather have something which is already written up.
Let $X$ be a ...
- 156k
3
votes
1
answer
529
views
Weak Lefschetz theorem for Lef line bundles
I'm studying
M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772.
The premises are the following....
- 828
2
votes
1
answer
589
views
Are finite correspondances flat?
In Voevodsky&Mazza&Weibel's book on motivic cohomology they define "an elementary correspondance from X (Smooth connected scheme over $k$) to Y (separated scheme over $k$) as an irreducible ...
- 2,871
2
votes
0
answers
96
views
On the root numbers of quadruples of quadratic twists of elliptic curves
We got strong numerical evidence for the root numbers and analytic ranks
of quadruples of elliptic curves over the rationals.
Related to this question.
Let $k,k_1,k_2$ be squarefree pairwise coprime ...
- 25.4k
2
votes
1
answer
211
views
Splitting of composition of trace and counit in derived setting
Let $X,Y$ be varieties (separated of finite type schemes) over base field $k$, $\mathcal{F}$ be constructible sheaf on $Y_{\mathrm{et}}$ and assume that we have a finite morphism $f: X \to Y$, which ...
- 5,946
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
2
votes
0
answers
255
views
For a nilpotent matrix A, are the cardinalities of sets: 1) B: commute with A, 2) B: anticommute with A, 3) B: q-commute with A — the same?
Let us work over finite fields $F_{p^k}$. Simulations seems to indicate:
Question 1: Consider a nilpotent matrix $A$, consider the set of all matrices $B$, such that $AB-qBA=0$, then cardinality of ...
- 24.9k
2
votes
0
answers
2k
views
What's the best reference for Abelian varieties?
I am curious about learning about Abelian varieties, specifically how they are in some ways generalizations of elliptic curves.
I know of the two sources: https://www.jmilne.org/math/CourseNotes/AV....
- 101
2
votes
2
answers
834
views
Shimura datum of family of fake elliptic curves
Suppose we have a PEL type $(H,\phi ,*;T,O,V)$ where H is a rational nonsplit quaternion algebra, $\phi$is an embedding of Q-algebra $\phi : H-->M(2,R)$, and * is a positive anti involution of H; ...
- 709
2
votes
1
answer
223
views
twisted cubic in a smooth hyperplane section of a cubic threefold
Let $X$ be a complex smooth cubic threefold and $C$ be a smooth twisted cubic, then $C\subset Y\subset X$ for a unique cubic surface $Y$ in $X$ (or equivalently a hyperplane section of $X$). When $Y$ ...
anon
2
votes
1
answer
563
views
Proposition 1.5 in Mumford's Geometric Invariant Theory
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\pr{pr}$I have some problems to understand the proof of Proposition 1.5 from Mumford's ...
- 5,946
2
votes
1
answer
232
views
Equivalence of sequences of blowups of $\mathbb{P}^3$
Let $[x_1,x_2,x_3,x_4]$ be coordinates of $\mathbb{P}^3$ and $Z\subset \mathbb{P}^3$ the subscheme given by the ideal $$I_Z=(x_1,x_2,x_3^2) \subset \mathbb{C}[x_1,x_2,x_3,x_4]$$ i.e. $Z$ is a double ...
- 1,343
2
votes
3
answers
917
views
Are the arithmetic genera of Cohen-Macaulay curves in a fixed homology class bounded?
Let X be a smooth projective variety over the complex numbers. Recall that a Cohen-Macaulay curve is a one-dimensional closed subscheme without embedded or isolated points (fat components are allowed)....
- 2,218
2
votes
1
answer
773
views
Is there an Abelian surface such that every effective divisor is ample? (Together with a boil down version to a question in Complex Lie group theory)
The Nakai-Moishezon criterion states that a line bundle $L$ over a surface $X$ is ample iff $L \cdot L > 0$ and $L \cdot C > 0$ for every curve $C$.
We can use this criterion to check that if ...
- 445
2
votes
1
answer
418
views
Sheaf cohomology and torsion
Let $X=Spec(R)$ be an affine scheme. Let $Y$ be a closed subset of $X$ and denote by $U$ its complement. Assume $U$ is quasicompact. Then $U= \cup_{i=1}^{n} D(f_i)$, where $f_i \in R$.
Denote the ...
- 23
2
votes
1
answer
228
views
Bounds on the coin-flipping degree
Let $p(\lambda)$ be a polynomial that maps the closed unit interval to itself and satisfies $0\lt p(\lambda)\lt 1$ whenever $0\lt\lambda\lt 1$.
The polynomial can be written in power form:
$$p(\lambda)...
- 697
2
votes
1
answer
642
views
How to show that the intersection of two certain affine varieties is reduced?
$\DeclareMathOperator\codim{codim}$Let $X=V(I)$, $Y=V(J)$ be two affine varieties. I'd like to know possibile strategies to understand when their schematic intersection, i.e. $X\cap Y=V(I+J)$, is ...
- 165
2
votes
0
answers
136
views
Question about the proof of the Duistermaat-VanDerKallen-Theorem, concerning the meaning of intersecting a chain with a set
On page 228 of the paper "Constant terms in powers of a Laurent polynomial" (by J.J. Duistermaat and Wilberd van der Kallen) a 'chain' $\Delta_\tau$ is defined as $f^{-1}([0,\tau])\cap \...
- 360
2
votes
1
answer
239
views
How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?
Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By ...
- 9,019
2
votes
0
answers
211
views
Lifting a morphism along quotient of a group action
Let $X$ and $Y$ be complex projective varieties. Assume there is a finite group $G$ acting on $Y$ and we denote the quotient projective variety by $Y/G$. We have a morphism of $\mathcal{Hom}$-schemes ...
- 5,901
2
votes
2
answers
321
views
A graphic representation of classical unitals on 28 points
I would like to understand the geometry of the classical unitals.
They are block designs containing $q^3+1$ points and whose blocks have cardinality $q+1$, where $q$ is a prime power. For $q=2$ (if I ...
- 42k
2
votes
1
answer
512
views
normalization of a stack
Hi,
how is it defined the normalization of an algebraic stack $A$ inside another algebraic stack $B$. If you do not want to write the answer could you give to me some reference?
Thank you
- 31
2
votes
0
answers
93
views
Morphism of varieties defined by the greatest common divisor
I'm reading the paper On the Chow Bunches for Different Projective Embeddings of a Complete Variety, Hoyt (1966). In this paper the author wants prove that the Chow monoind $\mathcal{C}_p(X)=\...
- 1,252
2
votes
1
answer
928
views
Paralel bezier curve
If I have a cubic Bezier curve specified by two endpoints and two control points, how can I find an offset curve which is "parallel" to the original at some given distance, after i have determined the ...
- 123
2
votes
1
answer
310
views
Lüroth theorem for $k \subset k(f,g) \subseteq k(x)$
Let $k$ be a field of characteristic zero (I do not mind to assume that $k=\mathbb{C}$, if things are easier in this case).
Lüroth theorem says that a field $L$, $k \subset L \subset k(x)$ containing ...
- 2,837
2
votes
1
answer
106
views
The commutativity of minimal extension and direct image by blowing-down
Let $X$ be a sooth algebraic variety over $\mathbb{C}$.
Let us assume that there exists the commutative diagram
$\require{AMScd}$
\begin{CD}
U @>{i}>> \hat{X}\\
@| @VV{\phi}V\\
U @>{j}>&...
- 111
2
votes
0
answers
134
views
Chebyshev-like Problem for Plucker Coordinates
$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}$
Let $n=2d+1$ be an odd integer, let $Gr(2,n)$ denote the Grassmmanian over $...
- 181
2
votes
1
answer
307
views
F-curves and divisors on $\overline{\mathcal{M}}_{g,n}$
It is conjectured that a divisor on $\overline{\mathcal{M}}_{g,n}$ would be ample if $D\cdot C >0$ for all F-curves $C\subset \overline{\mathcal{M}}_{g,n}$. Does the intersection degree with all F-...
- 3,779
2
votes
1
answer
180
views
Locally toric resolutions of compactifications
Suppose $U$ is a smooth, open $n$-dimensional variety over $\mathbb{C}.$ Say $X, X'$ are two proper normal-crossings compactifications of $U$. Call a map $m: X'\to X$ a modification if it is an ...
- 7,962
2
votes
0
answers
208
views
Sylvester-Gallai-type theorem for quadratic polynomials
Let $F_1, F_2$ and $F_3$ be finites sets of irreducible polynomials in $\mathbb{C}[x_0, \ldots, x_n]$ of degree at most $2$ such that $F_1 \cap F_2 \cap F_3 = \varnothing$ and for every $Q_1, Q_2$ ...
- 2,576
2
votes
1
answer
429
views
Classifying spaces and Brown's representability theorem
Let $G\text{-}PF(X)$ be the set of isomorphism classes of principal topological fibrations over the space $X$ with structural group $G$, and $G\text{-}PF_{cw} : hCW \to Set$ the contravariant functor $...
- 1,346
2
votes
1
answer
736
views
A question on nested Hilbert scheme
Let $H_1, H_2$ be two Hilbert schemes parametrizing subschemes in $\mathbb{P}^{n_1}, \mathbb{P}^{n_2}$ with Hilbert polynomials $P_1, P_2$, respectively. Given a pair $(Z_1, Z_2)$ of subschemes in $\...
- 2,032
2
votes
2
answers
1k
views
Injective resolution for right derived functor
This question is base on my previous question, and I repeat it here:
Suppose $X$ is a projective variety and $D^{+}(X)$ is the derived cateogry of bounded below complexes of sheaf of $\mathcal{O}_X$-...
- 3,472
2
votes
1
answer
591
views
commutative diagram with Yoneda pairing, Weil pairing and edge morphism
Why does the following diagram commute?$\require{AMScd}$
\begin{CD}
H^0(X,\mathscr{A}) \times \mathrm{Ext}^2_X(\mathscr{A},\mu_{\ell^n}) @>>> H^2(X,\mu_{\ell^n}) \\ @VVV @| \\
H^1(...
user19475
2
votes
2
answers
1k
views
Uniqueness on square root of complex Line Bundle
Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
user21574
2
votes
2
answers
521
views
spin bundle vs. hodge bundle
Let $\overline{\mathcal{M}}^r_{g,n}$ the space of $n$ pointed $r$-stable curves of genus $g$, endowed with a root of the canonical sheaf, and let $\mathcal{C} \to \overline{\mathcal{M}}^r_{g,n}$ be ...
- 3,779
2
votes
0
answers
263
views
Chevalley groups over $k[t]/t^n$
This question is motivated partly by a recent question on Chevalley groups over arbitrary commutative rings (and see also this older question). The answers to that question point to a large and ...
- 3,637
2
votes
1
answer
285
views
How to show $\dim_{\mathbb{A}_{\mathbb{R}}^n} V= \dim_{\mathbb{A}_{\mathbb{C}}^n} V$?
Suppose $V$ is an affine algebraic set defined by real polynomials.
Let $\mathbb{A}_{\mathbb{R}}^n$ be $\mathbb{R}^n$ endowed with Zariski topology where the closed sets are algebraic sets (in $\...
- 3,625
2
votes
0
answers
261
views
Codimension restrictions on intersections
This is a question I stumbled across earlier this week. I see a similar one has been asked here.
Suppose I have a smooth quasi projective variety $X$ over a field $K$, and I call $\text{Chow}^r(X\...
user113452
2
votes
3
answers
341
views
Linear homogenous polynomials that generates one quadratic polynomial
Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f$ be a homogenous quadratic polynomial of degree $2$.
Assume that for every $i$ and ...
- 2,576
2
votes
1
answer
296
views
Can we have "tropical polynomials" with arbitrary real powers?
I am referring to the definition of a tropical polynomial given on page 8 (top of section 3) here in this review, https://arxiv.org/pdf/math/0306366.pdf. I understand that here the notion of a ...
- 2,246