algebraic geometry – Meaning of superscript on $times$ for $G$-spaces

I’ve seen this notation a few times, and I can’t find any documentation of its meaning or basic properties (I have no idea what to search). Given an algebraic group $G$, a right $G$-torsor $mathcal{X} to X$, and a left $G$-space $Y$, how is the space $mathcal{X} times^G Y$ defined, and what is it called? When is it a $G$-torsor? Does the side of the $G$-action on $mathcal{X}$ or $Y$ matter? Is the notation symmetric? Are the conditions I’m imposing necessary, or am I missing some?

An example of its use (from Richarz, “A new approach to the geometric Satake equivalence”; note that $mathrm{Gr}_G$ is a right $fpqc$-quotient $mathrm{Gr}_G := LG/L^+G$):

Consider the following diagram of ind-schemes
$$
mathrm{Gr}_G times mathrm{Gr}_G overset{p}{leftarrow} LG times mathrm{Gr}_G overset{q}{to} LG times^{L^+G} mathrm{Gr}_G overset{m}{to} mathrm{Gr}_G.
$$

Here $p$ (resp. $q$) is a right $L^+G$-torsor with respect to the $L^+G$-action on the left factor (resp. the diagonal action). The $LG$-action on $mathrm{Gr}_G$ factors through $q$, giving rise to the morphism $m$.

If I want to take this as a definition (quotient a Cartesian product of two $G$-sets by the diagonal action), that still doesn’t answer all of my questions, and leaves me wondering what kind of quotient I should be using.

algebraic geometry – Chern classes of restrictions

Let $mathcal{F}$ be a vector bundle on some complex projective variety $X$ and let $Dsubseteq X$ be a divisor.
Is there any nice description of the Chern classes $c_i(mathcal{F}vert_D)$ of the restriction $mathcal{F}vert_D$ of $mathcal{F}$ as elements of the Chow ring of $D$ in terms of the Chern classes of $mathcal{F}$, $D$ and the inclusion map $i:Dhookrightarrow X$?

This might be a completely trivial question, but at the moment I am completely confused about the appearing notions, so any help would be appreciated.

ag.algebraic geometry – Distribution of rational points in the real locus of a planar algebraic curve

Let $C$ be a smooth projective geometrically connected curve over $mathbb{Q}$. Assume that $g(C)=3$ and that $C$ is not hyperelliptic. Then the canonical sheaf defines a closed immersion $Ctomathbb{P}^2_{mathbb{Q}}$. Tensoring with $mathbb{R}$ and analytifying we get an isotopy class of a nullhomologous one-dimensional submanifold of $mathbb{R}P^2$ that is either

  • empty
  • one oval
  • two ovals not encircling each other
  • one oval surrounded by another
  • three ovals not encircling each other
  • four ovals not encircling each other.

By Faltings’s theorem $C$ has finitely many rational points. In fact assuming the weak Lang conjecture there is an absolute bound on how many rational points a curve of fixed genus may have. Has the distribution of the rational points with respect to the real locus been studied?

A similar question has been asked for elliptic curves.

geometry – Ham sandwich theorem for other ratios than 1:1?

The Ham Sandwich theorem Let $A_1,dots,A_n$ be compact sets in $mathbb{R}^n$. Then there exists a hyperplane that bisects each $A_i$ simultaneusly into two pieces of equal area measure.

A possible generalisation comes to mind:

a) Given $0<r<1$ can we in general find a hyperplane that divides each $A_i$ into two pieces with the given ratio between them.

This should be false (it is an exercise in “Using the Borsuk-Ulam lemma” by Matousek). However I haven’t been able to come up with a counterexample. How would I show this is false for all $rne 1/2$?

We may also ask about further generalisations:

b) What about specifying ratios $r_1,dots, r_n$ for each set?

A counterexample here would be take all the $A_i$ to be the same set. What if we add the requirement that the $A_i$‘s are disjoint and “far apart” from each other, say with distance larger than the sets diameter? In this case it seems we should at least be able to specify one of the ratios to to be arbitrary.

Consider as an example the the ratios $(r_1,r_2) =(1/3,1/2)$ in $mathbb{R}^2$. To get the first ratio we add a compact set $B$ of area $A_1/3$ to $A$, creating a new set $A’_1 = A_1 cup B$. By placing $B$ sufficiently far away in a direction such that a line can’t intersect $A_1,A_2,B$ simultaneusly, the 1:1 bisection of $A’_1$ should give the desired split of $A_1$. Is this reasoning correct?

ag.algebraic geometry – A New Way the Area of Trapezium by Piyush Goel

Lot of mathematicians have proved Pythagoras theorem in their own ways. If you google it you will indeed found hundred of ways.

Meanwhile I was also sure that maybe one day I could find something new out of this incredible Pythagoras theorem and Recently I got something which I would like to share with you.

To Prove: Deriving the equation of area of trapezium using Arcs

Proof: There is a triangle ABC with sides a b and c as shown in the figure.

Now,

Area of ∆ BCEG = Area of ∆ BDC +Area of ⌂ DCEF + Area of ∆ EFG

c^2=ac/2+ Area of ⌂ DCEF + (c-b) c/2

(2c^2– ac –c^2+ bc )/2=Area of ⌂ DCEF

(c^2– ac+ bc )/2=Area of ⌂ DCEF

c(c– a+ b)/2=Area of ⌂ DCEF

Area of ⌂ DCEF=BC(DE+CF)/2. A New Way the Area of Trapezium by Piyush Goel
A New way, The Area of Trapezium
Copyrighted©PiyushGoel.

ag.algebraic geometry – Stalk of motivic homotopy sheaves

In contrast to “classical” homotopy theory, in the motivic homotopy theory, we don’t have homotopy group but rather homotopy sheaves in the Nisnevich topology, which is associated to the presheaf
$$pi_n^{mathbb{A}^1}(mathcal{X}):Sm_krightarrow text{Ab},quad Umapsto (S^{i,j}wedge U_+,mathcal{X})_{mathbb{A}^1}.$$
Morphisms of spaces $mathcal{X}rightarrow mathcal{Y}$ induce morphisms of homotopy sheaves $pi_n^{mathbb{A}^1}(mathcal{X})rightarrow pi_n^{mathbb{A}^1}(mathcal{Y})$. The Nisnevich topology has enough points,and thus an isomorphism of homotopy sheaves can be detected on stalks. So a natural question is what is the stalk of the homotopy sheaves? Is it enough to evalute $pi_n^{mathbb{A}^1}(mathcal{X})$ on the spectra of local henselian rings?

dg.differential geometry – Regularity of a singular Kaehler Einstein metric

On a manifold $X$ of general type i.e. $X$ is projective and $c_1(K_{X})$ semiample. One can construct a singular Kaehler Einstein metric $omega_{infty}$ in $-c_1(X)$. In particular, $omega_{infty}$ can be constructed as the limit of a normalized Kaehler Ricci flow, and in this case the potential $varphi$ of $omega_{infty}$ is shown to be $L^{infty}(X)$. I wonder whether it is possible to say something about $|nabla varphi|$ and higher derivatives of $varphi$.

dg.differential geometry – Every immersion can be deformed to have only transverse self-intersections

I asked this question some time ago in MSE, but obtained no answer. Maybe this is the right place to post it.

Let $f : M^n to overline{M}^{n+k}$ be an immersion between smooth manifolds. Is it true that there exists a smooth map $F : M times (0,1) to overline{M}$ such that the following conditions hold?

  1. $F_0 = f$;
  2. $F_t : M to overline{M}$ is an immersion for every $t in (0,1)$;
  3. $F_1(M)$ has only transverse self-intersections.

(Here, $F_t(p) = F(p,t)$ for every $(p,t) in M times (0,1)$.)

If this does not hold in this full generality, is it true for hypersurfaces ($k=1$)? For $overline{M} = mathbb{R}^{n+k}$?

ag.algebraic geometry – Hilbert scheme with essential generators

Let $H$ be the Hilbert scheme of finite flat lci subschemes in $mathbb{A}^n_k$ of degree $d$, where $k$ is some field. In other words the set of morphisms from an affine $k$-scheme $Spec(A)$ to $H$ consists of surjections $A(t_1, dots, t_n) to B$ (up to the evident notion of isomorphism) with kernel generated by $n$ elements, such that $A to B$ is finite locally free of degree d. (One knows that $H$ is smooth of dimension $nd$.) Let $U subset H$ denote the subfunctor consisting of those maps $A(t_1, dots, t_n) to B$ such that $A(t_1, dots, t_{n-1}) to B$ is surjective (i.e. the last generator is unnecessary). I believe this defines an open subscheme of $H$ (essentially by Nakayama’s lemma).

Here is my question: Write $Z subset H$ for the (reduced) closed complement of $U$ in $X$. What can be said about $Z$? In particular, what is its codimension?

Regarding the title: Naively, $Z$ consists of those maps $A(t_1, dots, t_n) to B$ such that $A(t_1, dots, t_{n-1}) to B$ is not surjective. In other words, the last generator is essential.

ag.algebraic geometry – tame ramification and separability of the reduction

Let $K$ a local field, with ring of integer $mathcal{O}_K$ or $mathcal{O}$ ,of uniformizer $pi$, residue field $k$ of caracteristic $p$.

Let $varphi:mathbb{P}^1_Ktomathbb{P}^1_K$ finite separable morphism with tame ramifications: all the order of vanishing are prime to $p$. Let $Phi:mathbb{P}^1_{mathcal{O}}tomathbb{P}^1_mathcal{O}$ an extension of $varphi$ to $mathbb{P}^1_mathcal{O}$ and let $overline{varphi}$ the reduction of $Phi$ in $k$ that is $overline{varphi}=Phitimes_mathcal{O}text{Id}_k:mathbb{P}^1_ktomathbb{P}^1_k$.
Concretely $varphi$ is associated with a fraction $pi^k P/Q$ with $P,Qinmathcal{O}(t)$,$Phi$ is $(pi^kP:Q)$ or $(P:pi^{-k}Q)$ (depending of the signe of $k$) and $overline{varphi}$ is the reduction of this fraction if it’s possible (constant otherwise).

Question: can we tell that if all the ramifications of $varphi$ are tame and $overline{varphi}$ is not constant (ie finite) then $varphi$ is a separable morphism? If not can you give me an example. If not, is there a criterion for the reduction to stay separable? Thanks!