ag.algebraic geometry – Special fiber of a reflexive sheaf over DVR

Let $f:X to mbox{Spec}(R)$ be a flat, projective morphism with $R$ a discrete valuation ring and the special and generic fibers of $f$ are normal and integral. I am looking for examples of rank $1$, reflexive sheaves on $X$ such that its restriction to the generic fiber is reflexive but its restriction to the special fiber $X_k$ is not a reflexive sheaf on $X_k$. Any hint/reference will be most welcome.

ag.algebraic geometry – Homotopy type of the affine Grassmannian and of the Beilinson-Drinfeld Grassmannian

The affine Grassmannian of a complex reductive group $G$ (for simplicity one can assume $G=GL_n$) admits the structure of a complex topological space. More precisely, the functor $$Xmapsto |X^{an}|$$ that associates to a scheme locally of finite type the underlying topological space of its analytification ((SGA1, XII, 1)) can be left-Kan-extended to a functor $PSh(Sch)to Top$, and therefore applied to the ind-scheme $Gr_G$.

Some topological properties of this space are known: its connected components are in bijection with the topological fundamental group $pi_1(G)$ ((Beilinson-Drinfel’d, Quantization of Hitchin’s integrable system and Hecke eigensheaves, Proposition 4.5.4)). The étale fundamental group (which is the profinite completion of the topological fundamental group) of each orbit in $Gr_G$ under the left action of $L^+G$ is trivial (Richarz, Affine Grassmannians and the Geometric Satake Equivalences, proof of Proposition 3.1).

Also, $|Gr_G^{an}|$ is known to be homotopy equivalent to a polynomial loop space ((Zhu, An introduction to the affine Grassmannian and the Geometric Satake Equivalence) and (Pressley/Segal, Loop Groups, Sec. 8.3)).

However, my knowledge essentially stops here. For example, the homotopy groups of this polynomial loop space are unknown to me.

Has anyone ever written or thought about the topology of the affine Grassmannian in further detail? Relevant questions to me are the following:

  1. What are the fundamental groups of the connected components of $|Gr_G^{an}|$?
  2. The cohomology of the affine Grassmannian is much more studied and used than its homotopic properties. Is there any chance that the connected components of $|Gr_G^{an}| satisfy conditions like “being a simple space”, which often allows homotopical issues to be translated into cohomological issues, by means of the homological Whitehead theorem?

Needless to say, the answer to such questions for the Beilinson-Drinfel’d Grassmannian will probably be more difficult. However, any ideas or references in this more sophisticated context would be highly appreciated.

Thank you in advance.

ag.algebraic geometry – Mumford’s definition of geometric quotient

Let $G$ be a group scheme over $S$, acting on an $S$-scheme $X$. In Mumford’s Geometric invariant theory, $S0.1$, Page 4, he defines (Definition 0.6) the notion of a geometric quotient. I won’t repeat the entire definition here, but one of the conditions for $phi : Xrightarrow Y$ to be a geometric quotient is that “$phi$ is surjective, and the image of $Gtimes_S Xrightarrow Xtimes_S X$ (given by $(g,x)mapsto (gx,x)$) is $Xtimes_Y X$.

He then remarks, “equivalently, the geometric fibers of $phi$ are precisely the orbits of the geometric points of $X$…”.

My main question is: What does he mean by “image”? I’m also curious how his remark is equivalent to saying that the image is $Xtimes_Y X$.

A similar question is here, where Torsten Wedhorn seems to imply that by “image”, he means the image as a subfunctor of the functor of points of $Y$. Namely, the map $phi : Xrightarrow Y$ induces a map of functors of points $h_phi : h_Xrightarrow h_Y$. There is a natural map $j : Xtimes_Y Xrightarrow Xtimes_S X$ and Torsten seems to imply that Mumford is requiring that for every $S$-scheme $T$, $h_phi(h_X(T)) = h_j(h_{Xtimes_Y X}(T))$ as subsets of $h_Y(T)$.

Is this what Mumford means? Is this a standard notion of “image”? (It doesn’t seem to explained anywhere before the definition) At first I thought it might be scheme theoretic image, but scheme theoretic images are always closed, and Mumford doesn’t seem to make any separatedness hypotheses (after all he’s talking about preschemes), so I don’t think $Xtimes_Y Xrightarrow Xtimes_S X$ needs to be a closed immersion.

The stacks project also gives a definition of geometric quotient for algebraic spaces, which essentially amounts to Mumford’s condition on geometric points. I’m wondering if Mumford had another concept in mind that ends up being equivalent to the condition on geometric points.

Also, if anyone has any suggestions for good modern references to the general theory of quotients by reductive groups (over $mathbb{Z}$), that would be welcome as well.

differential geometry – For the same curve,The critical point becomes a regular point under different parameter expressions.

When I say “the same curve”,I mean they have the same image in the $R^2$ plane.

When I say “a critical point of $mathbb{r}(t)$“,I mean “$mathbb{r}^{‘}(t_0)=mathbb{0}$“.

Let’s take a curve which looks like y=x as an example.

$$ left{
begin{aligned}
x & =t\
y & =t
end{aligned}
right.
$$

It is surely a regular point when t is equal to zero.

However,when it comes to
$$ left{
begin{aligned}
x & =t^3\
y & =t^3
end{aligned}
right.
$$

The point where t is equal to zero becomes a critical point!

I’m afraid that these two curves may not be the same curve by the defination in differential geometry,are they?

And I still have another question.I think the regularity of a point describes whether the curve is smooth at this point or not.But the latter go against my intuition:it is critical at the point t =0 by defination while it is obviously smooth everywhere because it looks like y=x!

ag.algebraic geometry – Algebras Morita equivalent with the Weyl Algebra and its smash products with a finite group

My question os motivated, naturally, by the problem of classifying symplectic reflection algebras up to Morita equivalence (a classical reference for rational Cherednik algebras is Y. Berest, P. Etingof, V. Ginzburg, “Morita equivalence of Cherednik algebras” MR2034924; the most up do date work in this subject I know of is I. Losev https://arxiv.org/abs/1704.05144);

and also the problem of understading rings of differential operators on irreducilbe affine varieties $X$ up to Morita equivalence (a nice discussion of this problem can be found in Y. Berest, G. Wilson, “Differential isomorphism and equivalence of algebraic varieties” MR2079372)

Given that:

(Question 1): What are the more general known conditions on a symplectic reflection algebra $H_{1,c}(V,Gamma)$ that imples it is Morita equivalent to $mathcal{D}(V) rtimes Gamma$?

(Question 2): What are the recent developments made in the study of equivalence of rings of differential operators up to Morita equivalence (and in particular Morita equivalent to the Weyl algebra) since Berest, Wilson (op. cit.)?

(Question 3): Etingof in “Cherednik and Hecke algebras of varieties with a finite group action” MR3734656 introduces more general versions of rational Cherednik algebras and discuss the possibility of extending the results in Y. Berest, O. Chalykh, Quasi-invariants of complex reflection groups MR 2801407 in this setting. So, being optmistic, one hipotetically could obtain results similar as those discussed in Berest, Etingof, Ginzburg (op. cit) regarding Morita equivalence of these generalized rational Cherednik algebras with smash products of rings with differential operatos with a finite groups. Has this line of inquiry lead to results relvant to this discussion so far?

(Question 4): This is totally unrelated to the previous questions. It is more of a very open question in ring theory: are there interesting simple Noetherian algebras, coming from another areas than those above, which are Morita equivalent to a Weyl algebra or a smash product of it with a finite group?

ag.algebraic geometry – Kan liftings and projective varieties

Regard the following two bicategories:

  • $operatorname{dg-mathcal{B}imod}$, with objects dg categories, and morphisms categories from $C$ to $D$ being the categories of $C$$D$-bimodules. Composition is given by the dg tensor product. Note that this might be considered as the bicategory of Chain-complex-enriched profunctors/ relators (see Coend Calculus), so the machinery developed in this paper should (as chain complexes form a Bénabou-cosmos) show that every 1-morphism in this category has both a left and a right adjoint, and Left Kan Liftings exist here
  • $mathcal{V}ar$, with objects smooth projective varieties and morphism categories from $X$ to $Y$ the derived categories $D^b (X times Y)$, as considered here. The 1-morphisms here encode kernels for Fourier-Mukai-Transformations. Using the Serre kernel, it is shown in that paper that every 1-morphism in this bicategory has a left and a right adjoint.

These bicategories are very closely related; it was shown e.g. by Orlov that if one takes the dg enhancements of the derived categories $D^b_{dg}(X)$, $D^b_{dg}(Y)$, then the dg functors between those are in one-to-one correspondence with the elements of $D^b(X times Y)$, to name just one similarity. Therefore, I would suspect that also the second category possesses Left Kan Liftings, but I am not sure how to construct them (as already the proof that the Serre kernel lets us construct adjoints there is very nontrivial).

Is this indeed true? How are they constructed? And most importantly (as from the fact that $mathcal{V}ar$ has adjoints, a lot can be deduced about Grothendieck duality etc.) do they also have interesting uses in the study of smooth projective varieties, or can their construction even be extended to more general schemes?

euclidean geometry – 3 chords inclined at $pi/3$ of a convex closed curve that intersect at their midpoint always exist

In “The penguin dictionary of curious and interesting geometry” David Wells mentions the following property of closed convex curves without a reference nor a proof.

“Given any closed convex curve, it is possible to find a point P, and three chords inclined at $pi/3$, such that P is the mid-point of all three”.

Can anyone redirect me to some literature or provide an argument for such a property?

ag.algebraic geometry – An analogue of Noether’s Problem for non-rational varieties

For the sake of simplicity, let $mathsf{k}$ be algebraically closed and of zero characteristic. Varieties are irreducible.

The (linear) Noether’s Problem (which goes back to the early 1910’s in Burnside’s and Noether’s work on invariants) is the following: let $V$ be a finite dimensional vector space, and $G subset GL(V)$ a finite group. When is the variety $V/G$ rational?.

Now this problem is intrinsically very nice and has a miriad of applications to moduli questions (cf. C. Böhning https://arxiv.org/abs/0904.0899; J-L. Colliot-Thélène, J-J. Sansuc, The rationality problem for fields of invariants under linear algebraic groups MR2348904 );
PI-algebras (E. Formanek, The polynomial identities and invariants of n×n matrices, MR1088481); the inverse Galois problem (D. Saltman, Groups acting on fields: Noether’s problem, MR810657); etc.
It is also interesting that the group actions that give a positive solution to Noether’s Problem are very rich in nature.

Some questions closely related to this Problem have arisen in my research, and I wonder if they are adressed in some paper, or anyone knows something about then. I thought about these questions, but I could only think of examples coming from elliptic curves (such as multiplication by $(n)$ map), but these are, to my taste, a little bit too ‘simple’.

(Q1): Let $X$ be a smooth projective curve of genus at least 1. What are the finite groups $G$ of $Aut_mathsf{k} , X$ such that $X/G$ is a smooth projective curves of the same genus?

(Q2): Let $X$ be a non-rational variety and $G$ a finite group of $Aut_mathsf{k} , X$. When $X/G$ is birationally equivalent to $X$ itself?

ag.algebraic geometry – Can $h^{1, 1}$ jump for smooth projective surfaces over $mathbb{Z}[1/N]$?

Let $N$ be a positive integer. Let $f:Xto S=mathrm{Spec}:mathbb{Z}(1/N)$ be a smooth projective morphism of relative dimension 2.

For a closed point $sin S$ denote the ring of Witt vectors over $k(s)$ by $R_s$. Assume that for all closed points $H^ibig(X_s, WOmega^j_{X_s}big)$ is a finitely generated $R_s$-module for all $i, jgeq 0$ and $H^i_{mathrm{crys}}(X_s/R_s)$ is a torsion-free $R_s$-module for all $igeq 0$. By Deligne-Illusie the Hodge-de Rham spectral sequence degenerates for all closed fibers so a theorem of Joshi implies that Hodge symmetry holds for all closed fibers.

Additionally assume that $R^1f_*mathcal{O}_X$ and $R^2f_*mathcal{O}_X$ are both locally free $mathcal{O}_S$-modules. Then $mathrm{dim}_{k(s)}H^ibig(X_s, Omega^j_{X_{s}/k(s)}big)$ can not vary with $s$ unless $i=j=1$. Is this Hodge number constant as well or is there a counterexample?

geometry – Find the area of the triangle knowing the length of perp bisector from side to circumcircle

Let $triangle ABC$ inscribed in circle with center $O$ and radius $r$. Let $D,E,F$ be the mid-point of side $BC,CA,AB,$ respectively then $OD,OE,OF$ meet the circumcircle at $L,M,N$ respectively. If $DL=a, EM=b, FN=c$ then find the area of the triangle.

Could someone help me with this? I approach using Pythagorean to find radius but end up with everything just equal to each other as it should be.