Tagged Questions
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
0
votes
0answers
30 views
Bertini theorem for big divisors and klt pairs
Let $X$ be a smooth projective variety and let $D$ be a big $\mathbb Q$-divisor on $X$. Assume that for $m$ large $|mD|$ has no fixed components. Is there a $\mathbb Q$-divisor $D'\equiv D$ so that ...
0
votes
0answers
87 views
Moving lemma for algebraic curves
Let $X$ be a smooth irreducible projective curve contained in $\mathbb{P}^3$ and $Y$ be another reduced but not necessarily irreducible curve in $\mathbb{P}^3$. Denote by $P$ the Hilbert polynomial of ...
1
vote
1answer
126 views
Blowing-up the Grassmannian at a point
Does anyone know what the blow-up of the Grassmannian at a point looks like? Consider $G=Gr(r,n)$ and $V\in G$. I want to understand more explicitly what $Bl_V(G)$ should mean.
Of course for affine ...
0
votes
0answers
51 views
A combinatorial question on ranks
Denote
$$\mathscr{C}[r]=\{Q\in\Bbb Z_{\geq 0,\leq 1}^{n\times n}:\mathsf{rk}(Q)= r\}.$$
$$\mathscr{D}[A,t]=\{B\in\mathscr{C}[\mathsf{rk}(A)]:\mathsf{dim}(col(A)\cap col(B))\geq t\}.$$
Given ...
1
vote
2answers
211 views
Tensor calculus on the frame bundle
Let $M$ be a manifold and let $g$ be a tensor on it, say for example a metric $g\in\Gamma(T^{\ast}M\otimes T^{\ast}M)$. I know how to perform any computation on $g$. For instance, taking its ...
-1
votes
0answers
51 views
Minimum rank non-negative matrix summations
Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$ of rank $r$.
What is minimum $k$ such that
$$\mathscr{A}[b,k]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(Q)\leq k\}$$ contains $R,S$ ...
4
votes
1answer
229 views
Is the ring of invariants Noetherian?
Let $R$ be a complete regular local ring whose residue field is perfect. Suppose that a finite group $G$ acts on $R$ by ring automorphisms in such a way that the induced action on the residue field is ...
7
votes
0answers
201 views
Can an abelian variety/Q have no points over Q_sol?
Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal ...
3
votes
1answer
159 views
Reference for Arakelov's theorem: $K^2_f=0$ iff $f$ is locally trivial
Let $f:X\longrightarrow B$ be a family of curves, with $f$ relatively minimal, over a fixed curve $B$ ($B$ is projective, irreducible and smooth). The fibration $f$ is said locally trivial if all ...
2
votes
0answers
180 views
When does a perverse sheaf occur in the decomposition theorem?
Suppose I am in the setting of the decomposition theorem, i.e., we have the decomposition of the direct image $f_*\mathbb Q_\ell$, where $f:X\to Y$ is proper. Then the direct image decomposes into a ...
0
votes
0answers
73 views
Regular embeddings of reductive groups
A regular embedding of a connected reductive linear algebraic group $G$ defined over $\mathbb{F}_q$ is a morphism $\varphi : G \rightarrow G'$ of algebraic groups which is a closed immersion where ...
-1
votes
0answers
50 views
Calculate the intersection numbers by a plane section [on hold]
This question is from the chapter A of Reid's note: Chapters on algebraic surfaces
Let X = X$_d$ $\subset$ P$^3$ be a nonsingular surface of degree d and
suppose that X has a plane section P ...
3
votes
0answers
111 views
Blowig-up a point in the singular locus
Let $X\subset\mathbb{P}^n$ be a variety singular along a smooth subvariety $Z\subset X$ of positive dimension. Let us assume that $X$ has ordinary singularities along $Z$. Now, let $\pi:Y\rightarrow ...
1
vote
1answer
94 views
Is there a formula for the intersection of projectivized lines inside a projectivized vector bundle?
Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex
one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E
(i.e. line bundles). Let
$$ n_1:= ...
1
vote
1answer
164 views
How does one compute the first Chern class of a Line bundle defined as the Kernel of a linear map?
Let $M$ and $N$ be compact complex manifolds of the same dimension ($m$) and
$\mu: M \rightarrow N$ a holomorphic map. Let $D \subset M$ be the subset of
points of $M$, where $d\mu|_p$ fails to be ...
0
votes
1answer
136 views
Function field of the Jacobian of genus 2 curve over $\mathbb{F}_q$
I have been trying to build the function field of the jacobian of a genus 2 smooth curve over a finite field, but I am having problems making it explicit, I need to work with another curve with points ...
0
votes
0answers
63 views
Pro-constructible subset of scheme intersects very dense subsets?
Let $X$ be a scheme, let $D$ be a very dense subset of $X$ and let $Y$ be a pro-constructible subset of $X$. Is it true that $Y \cap D \neq \emptyset$?
If $Y$ is just constructible, this is true.
...
0
votes
0answers
157 views
Multiplication Map, Is it invariant?
Let $\pi:X\rightarrow Z$ a double cover of an elliptic curve with genus $g\geq 3$. Choose a general rank 2 and degree -1 vector bundle $F$ on $Z$, let $E=\pi^*F$ and fix $x\in X$. The involution $i$ ...
1
vote
1answer
164 views
Relation between intersection and product of ideals
Let $C$ be a smooth projective (irreducible) curve in $\mathbb{P}^n$ for some $n$. Denote by $I_C$ the ideal of $C$. Let $g \in I_C\backslash I_{C}^2$, an irreducible element. Is it true that for any ...
0
votes
0answers
60 views
A simple question about a resolution of a conifers singularity
Let $X$ be a conifold defined by the equation $xy-zw=0$ in $\mathbb{C}^4$ and $\tilde{X}$ its crepant resolution, which is isomorphic to $\mathcal{O}_{\mathbb{P^1}}(-1)^{\oplus 2}$. Then there is a ...
1
vote
1answer
28 views
About convex combinations of real-stable multivariable complex polynomials
Say $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$ is a real stable multivariable polynomial on the variables $(z,w_1,w_2,...,w_n)$. (a "real-stable" polynomial is one which has no zeroes in the open ...
2
votes
1answer
210 views
What happens to the cohomology ring after a “flip-flop”?
I've been trying to understand what happens to the cohomology ring (say with coefficients in $\mathbb{R}$) of a smooth complex projective manifold after blowing up along a smooth complex submanifold. ...
2
votes
0answers
59 views
Higher genus Gromov-Witten potential
Is it known if the higher genus (gravitational) Gromov-Witten potential is split in a classical and quantum part like the genus 0 Gromov-Witten potential? If so, Could someone give a reference?
8
votes
2answers
239 views
Implicit Function Theorem on Singular Varieties
Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function
$$f\colon X \to Y $$
and assume that it is bijective at the level of ...
9
votes
2answers
359 views
Is there an analogue of the Tate (and Hodge) conjecture for varieties that are not proper smooth (i.e., the mixed case)?
Let $X/K$ be a variety (scheme of finite type, geometricaly integral) over a finitely generated field $K$. If it is smooth and proper, we can formulate the Tate conjecture, and if $\text{char}(K) = 0$ ...
7
votes
1answer
212 views
A proper smooth surface is projective
My question is a reference request for the following fact: if $k$ is a field and $X$ a proper smooth surface over $k$, then $X \rightarrow \mathrm{Spec}\, k$ is projective. Where is this well-known ...
5
votes
0answers
91 views
Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag variety G/B
This is a follow-up of this previous question below:
Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some ...
4
votes
0answers
138 views
Automorphisms of a quotient variety
Let $X$ be a variety, and $G\subset Aut(X)$ a subgroup of the automorphism group of $X$. Assume that the quotient $Y = X/G$ is a variety. Does there exist some simple relation between $Aut(X)$, $G$ ...
1
vote
0answers
128 views
About real roots of complex multivariable polynomials
Let $f(z,w_1,w_2,..,w_n)$ be a multivarible complex polynomial mapping $\mathbb{C}^{n+1} \rightarrow \mathbb{C}$ and it has all real coefficients. Assume that this is "real-stable" i.e it has no roots ...
-1
votes
0answers
184 views
What is the error here? [closed]
Let $X$ a curve of genus $g\geq 3$ with a double cover to an elliptic curve $Z$. Let $F$ be a rank $2$ and degree $1$ locally free sheaf on $Z$, and $G$ its pullback to X.
Then, by Serre duality ...
1
vote
0answers
64 views
Are all (graded) Artinian complete intersections like this?
I'm trying to prove some stuff (it's not important what) about (graded) Artinian complete intersections $R=\mathbb{C}[x_1,\ldots,x_n]/I$, where the $x_i$ have certain positive weights and where $I$ is ...
1
vote
0answers
67 views
“Exceptional components” of the exceptional divisor of a blow up
Assume we are blowing up an ideal $I$ on an affine variety $X$, let $E$ be the exceptional divisor, and $P$ be a (closed) point in $V$, the zero set of $I$. Is there any algorithm to check that $E$ ...
1
vote
0answers
66 views
Degree 2 curves on a degree d hypersurface in P^(2d+2)/3
One of the foundations of Gromov-Witten theory is the use (due to Kontsevich I think) of localization to calculate the number of degree $n$ curves on a general quintic 3-fold. When calculating the ...
6
votes
1answer
252 views
Find a polynomial not in any ideal generated by polynomials of total degree $o(n)$
Is there an explicit nontrivial (= not a constant) polynomial $p \in \mathbb{C}[x_1, \ldots, x_n]$ such that, for any ideal $I \not= \mathbb{C}[x_1, \ldots, x_n]$ generated by $f_1, f_2, \ldots, f_m$ ...
3
votes
1answer
117 views
reference for “curves over S are locally the base change of a curve over S' which is finite type over R”
So recently I heard someone claiming that if $X\rightarrow S$ is a smooth curve (not necessarily proper?) and $S$ is an arbitrary scheme over $\text{Spec }R$ (for $R$ sufficiently nice), then there is ...
3
votes
2answers
195 views
A question on the effective cone
Let $X$ be a projective variety and $G$ a finite group acting on $X$. We consider the quotient $\pi:X\rightarrow Y :=X/G$.
I'm interested in the relation between $Eff(X)$ and $Eff(Y)$. In ...
1
vote
0answers
59 views
Normal bundles of rational equivalent curves
Let $C_1, C_2$ be rationally equivalent curves in a smooth projective variety $P$. Let $$N_i: = \mathcal{H}om(I_{i}/I^2_{i}, \mathcal{O}_{C_i})$$ be the normal bundle of $C_i$, where $I_i$ is the ...
4
votes
1answer
146 views
liftings of principal bundles
I would like to know what structure has the category of liftings of a principal bundle. Let me be more precise.
Fix $k$ an algebraically closed field and $X$ a smooth projective variety over it (for ...
2
votes
0answers
71 views
Deformation with fixed ramification
Suppose that $f : X \to Y$ is a finite, surjective morphism of normal varieties. I want to know about the space of first-order deformations of $X$ over $Y$ with fixed ramification, i.e. the ...
0
votes
1answer
79 views
Intersection multiplicty and global sections
Let $X$ be a smooth projective variety, $V, W$ closed subschemes in $X$ such that $V \cap W$ is finitely many points. Let $\mathcal{L}$ be a line bundle on $X$. Is there any relation between ...
1
vote
1answer
217 views
Proof of “generic curve of genus at least 2 has no nontrivial maps to a positive genus curve”
I searched for it for a long time, but it seems that everybody is taking this for granted and does not bother to point out a proof. Would it be possible that someone points me to a proof or makes me ...
5
votes
1answer
358 views
Structure of the automorphism group of a Riemann surface
I was wondering if anything is known about the possible structure of $\mathrm{Aut}(S)$ for a Riemann surface $S$. More precisely, are there known obstructions for a finite group $G$ to be such an ...
5
votes
0answers
151 views
Adjunction map for projective surfaces
Before stating my question, let me recall (part of) the classical result on the adjunction map for complex projective surfaces, due in this modern form to Beltrametti and Sommese:
Adjunction ...
7
votes
0answers
79 views
Weierstrass division theorem for henselian rings
Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...
0
votes
1answer
146 views
Hyperelliptic curve of genus 2 over R
I know that the points of an elliptic curve over $\mathbb{Q}$, $\mathbb{R}$ or other field $K$ form a group, particularly the most common example to explain the naive way is with this curve ...
0
votes
0answers
52 views
constant of functional equation of zeta function
Let $C$ be a smooth projective curve, of geometric genus $g$, over a finite field $\mathbb{F}_p$ and consider the zeta function $$
Z(C/\mathbb{F}_p, t)=\exp(\sum_{n=1}^{\infty} |C(\mathbb{F}_{q^n})| ...
-2
votes
0answers
9 views
Finding coordinates of pyramid with known base, and known angles for apex [migrated]
I have a regular pyramid, where I know the 3d coordinates of all points on the base, and I know all of the angles associated with the apex. I'm wondering if there's a known method to determine the ...
2
votes
0answers
153 views
Fixed points of self maps
Given $m$ points on $S^n$, is there an explicit polynomial self $1-1$ map of minimum degree $f:S^n\rightarrow S^n$ that fixes only these $m$ points? Can we say something about symmetry group of $f$ if ...
0
votes
0answers
86 views
A birational compatification problem
Let $P_1$ be a projective variety over $\mathbb{C}$, and $Y_1 \subseteq P_1$ be a codimension $1$ (irreducible) subvariety. Suppose there is a blowdown morphism $Y_1 \to Y_2$.
Then can I find a ...
2
votes
1answer
285 views
Exactness on rational points of algebraic groups
Let $k$ be a finite extension of the p-adic number field $Q_p$ and G be a connected algebraic (not affine) group over $k$. It is well-known (see e.g. [1] Proposition 3.1) that G decomposes as
...
