ag.algebraic geometry – Local rings $R subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular

Let $R subseteq S$ be local rings with maximal ideals $m_R$ and $m_S$.
Assume that:

(1) $R$ and $S$ are (Noetherian) integral domains.

(2) $dim(R)=dim(S) < infty$, where $dim$ is the Krull dimension.

(3) $R$ is regular (hence a UFD).

(4) $S$ is Cohen-Macaulay.

(5) $R subseteq S$ is simple, namely, $S=R(w)$ for some $w in S$.

(6) $R subseteq S$ is free.

(7) $R subseteq S$ is integral, namely, every $s in S$ satisfies a monic polynomial over $R$.

(8) $m_RS=m_S$, namely, the extension of $m_R$ to $S$ is $m_S$.

(9) It is not known whether the fields of fractions of $R$ and $S$, $Q(R)$ and $Q(S)$, are equal or not.

(10) It is not known if $R subseteq S$ is separable or not.

Remark: It is known that if a (commutative) integral domains ring extension $A subseteq B$ is integral+flat, then it is faithfully flat, and if also $Q(A)=Q(B)$, then $A=B$.
This is why I did not want to assume that $Q(R)=Q(S)$, since in this case $R=S$ immediately.

Question:
Is it true that, assuming (1)(10) imply that $S$ is regular or $R=S$?

Example:
$R=mathbb{C}(x(x-1))_{x(x-1)}$ and $S=mathbb{C}(x)_{(x)}$,
with $R neq S$ and $S$ is regular.

Non-example: $R=mathbb{C}(x^2)_{(x^2)}$ and $S=mathbb{C}(x^2,x^3)_{(x^2,x^3)}$, but condition (8) is not satisfied.

Relevant questions, for example: a, b, c, d.

Thank you very much! I have asked the above question here, with no comments (yet).

geometry – How to find the composite of an inverse function?

Consider the two parameters σ(u,v), $σ(u ̃,v ̃)$ of $x^2$+$y^2$+$z^2$=1, where x,y,z $>$ 0. Given σ(u,v) = $(u,sqrt{-u^2-v^2},v)$ and $σ ̃(u ̃, v ̃)$ = $(sqrt{1-u ̃^2-v ̃^2, u ̃, v ̃})$, where u,v,$u ̃,v ̃ in U.$ U = {(x,y): $x>0$ and $v>0$ and $x^2+y^2<1$.
Find $phi$= $σ ̃^{-1} ◦ σ$.

ag.algebraic geometry – Vanishing of intermediate cohomology for a multiple of a divisor

Let $S subset mathbb P^3$ be a smooth projective surface (over complex numbers). Let $C$ be a smooth hyperplane section. Let $Delta$ be a non-zero effective divisor on $S$ such that $h^1(mathcal O_S(nC+Delta))=0, h^1(mathcal O_S(nC-Delta))=0$ for all $n in mathbb Z$. Then my question is the following :

In this situation can we say that: $h^1(mathcal O_S(m Delta))=0$ for $m geq 2$? Can we impose any condition so that this happens?

Any help from anyone is welcome.

ag.algebraic geometry – Algebraic properties of geodesics

This is a question related to my last post. I will use the same definition here.

A complete smooth manifold $M$ with an affine connection $nabla$ is said to have an algebraic model of dimension $n$ if there exists a smooth immersion $sigma:M rightarrow Bbb{R}^n$ such that each image of geodesics on $M$ with respect to $nabla$ is either sub-algebraic or improper. A sub-algebraic set is a subset of $Bbb{R}^n$ defined by the common zeros and positive or non-negative parts of finitely many polynomials. (For example, half circles and toroidal handles are sub-algebraic sets.) In this sense, all simply connected hypaerbolic spaces have algebraic models (e.g. upper half space, Poincare’s unit $n$-ball). Note that I do not require $sigma$ to be algebraic — the usual hyperbolic cases are apparently not algebraic.

The first question arises naturally on how ‘algebraic’ $sigma(M)$ really is. I conjectured that if $M$ has an algebraic model $sigma$ then $sigma(M)$ is sub-algebraic — more specifically, the ‘infinite boundary’ of $sigma(M)$ is an algebraic set. It seems a counterexample is not easy to construct.

The second question concerns about a certain type of manifolds, namely the 3-manifolds with geometric structures in Thurston’s sense. I want to know if there is an algebraic model of at least 1 manifold of each of the 8 types. And as they have Lie groups as transition groups, I also conjectured that every geometric 3-manifold has an algebraic model. (The corresponding 8 Lie groups all seem to have algebraic models.) I am looking forward to any reference articles on this problem.

differential geometry – Nowhere-vanishing form $omega$ on $S^1.$

This is an example(19.8 and 17.15) from Intro to manifolds by Tu.

Let $S^1$ be the unit circle defined by $x^2+y^2=1$ in $mathbb{R}^2$. The 1-form $dx$ restricts from $mathbb{R}^2$ to a $1-form$ on $S^1.$ At each point $pin S^1$, the domain of $(dxmid_{S^1})_p$ is $T_p(S^1)$ instead of $T_p(mathbb{R}^2)$: $(dxmid_{S_1})_p:T_p(S^1)rightarrowmathbb{R}$. At $p=(1, 0)$, a basis for the tangent space $T_p(S^1)$ is $partial/partial y$. Since $(dx)_p(frac{partial}{partial y})=0,$ we see that although $dx$ is nowhere-vanishing $1-$form on $mathbb{R}^2$, it vanishes at $(1, 0)$, when restricted on $S^1.$ Define a $1-form$ $omega$ on $S^1$ by $omega=frac{dy}{x}$ on $U_x$ and $omega=-frac{dx}{y}$ on $U_y$ where $U_x={(x,y)in S^1mid xneq 0}$ and $U_y={(x,y)in S^1mid yneq 0}$.

I understand $omega$ is $C^infty$ and nowhere-vanishing. I want to understand why $omega$ on $S^1$ is the form $-ydx+xdy$ of Example below:

Example 17.15 (A 1-form on the circle). The velocity vector field of the unit circle $c(t)=(x,y)=(cos t, sin t)$ in $mathbb{R}^2$ is $c'(t)=(-sin t, cos t)=(-y, x)$. Thus $X=-yfrac{partial}{partial x}+xfrac{partial}{partial y}$ is a $C^infty$ vector field on the unit circle $S^1$. What this notation means is that if $x,y$ are the standard coordinates on $mathbb{R^2}$ and $i:S^1hookrightarrowmathbb{R}^2$ is the inclusion map, then at a point $p=(x,y)in S^1$, one has $i_ast X_p=-ypartial/partial xmid_p+xpartial/partial ymid_p$, where $partial/partial xmid_p$ and $partial/partial ymid_p$ are tangent vectors at $p$ in $mathbb{R}^2$. Then if $omega=-ydx+xdy$ on $S^1$, then $omega(X)equiv 1.$

ag.algebraic geometry – Automorphism of a stack morphism

Let $X$ be an algebraic stack and let $f: S to X$ be a smooth covering of $X$ by a scheme $S$.

Motivation: Forgetting about stacks for a moment and going back to covering spaces: Given a covering map $f: Z to Y$, with $Z$ connected and $Y$ locally connected, then $operatorname{Aut}(Z/Y)$ acts properly and discontinuously on $Z$. Moreover, if $operatorname{Aut}(Z/Y)$ acts transitively on a fiber of $p in Y$, then the covering is a $G=operatorname{Aut}(Z/Y)$-covering in the sense that $f: Z to Y cong Z/operatorname{Aut}(Z/Y)$ is a quotient map.

I am interested in making an analogous statement in the case of a smooth cover $f:S to X$ of an algebraic stack $X$. (Of course, dropping words like properly and discontinuously and keeping in mind that $f$ is not finite ‘etale and thus not a covering map in the above sense).

In particular, I want to describe $operatorname{Aut}(S/X)$. The “elements” of $operatorname{Aut}(S/X)$ are maps $phi: S to S$ such that $f circ phi =f$.
On the other hand, $f: S to X$ can be identified with a unique object $s in X(S)$ (up to $2$-isomorphism?) by the 2-Yoneda lemma and so it seems like $operatorname{Aut}(S/X)$ should have an interpretation in terms of the groupoid of maps $s to s$ lying over a given $phi: S to S$.

That is all very abstract, so let us just suppose that the elements of $X(S)$ have some geometric interpretation, for example, the object $s in X(S)$ is a family of genus g curves $C$ over a scheme $S$.

(1). Is there an interpretation of the groupoid $s to s$ lying over $phi: S to S$ in terms of a group automorphisms of $C$ over $S$?

(2). Moreover, how would this group act on a “fiber” $S times_{X,g} T$ (a sheaf on the category $Sch/S times T$?) over $g: T to X$?

ag.algebraic geometry – Poincarè dual of the exceptional divisor in the Kaehler setting

Let $(mathbb{C}^n,omega)=:(X,omega)$ be the complex $n$-space endowed with the standard Kahler form $isum_{j}dz_jwedge dbar{z}_j$ and let $pin X$ to be the origin. Consider the Blow-up of $X$ at $p$ denoted by $tilde{X}:=Bl_p(X)$. Then this is a Kahler manifold of complex dimension $n$ with Kahler form given by begin{equation*}tilde{omega}:=kcdotpi^*(omega)+alphaend{equation*}where $pi: tilde{X}to X$ is the blow-up map, $alpha$ is a representative of $c_1(mathcal{O}(-E))$ (here we are using that the exceptional divisor $E$ is a smooth Cartier divisor in $tilde{X}$) and $k$ is a positive constant taken to ensure that $tilde{omega}$ is positive as $(1,1)$-form. One can prove that $E$ (which is a copy of $mathbb{P}^{n-1}$ in $tilde{X}$) intersects itself negatively. From this follows that $E$ is the unique smooth complex representative of its homology class in $tilde{X}$ (if not, just perturbed $E$ a bit in its same homology class, then the two submanifolds will have positive intersection). In particular we have an isomorphism (induced by Poincarè duality) begin{equation*}PD: H_{2n-2}(tilde{X},mathbb{R})overset{sim}{longrightarrow} H^2(tilde{X},mathbb{R}) \ (E)mapsto PD(E) end{equation*}We can apply Poincarè duality since $tilde{X}$ can be identified with the total space of $mathcal{O}(-1)$ over $mathbb{P}^{n-1}$ which is compact (am I right?)

Question 1

Why is the form $eta$ defined by $(eta)=-PD(E)$ a Kahler form on $tilde{X}$?

Well $eta$ is obvious closed by definition. Then it seems to me that $eta$ is positive since it’s the opposite of $PD(E)$ and $E.E=-1$ but I am not able to give a rigorous proof. Moreover I really can’t see why it should be a $(1,1)$-form

Question 2

Do such Kahler forms have zero constant scalar curvature?

Perhaps this is a very hard question but if anyone can help I appreciate.

mg.metric geometry – Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and fundamental group which is not relatively hyperbolic?

It is well known that a geometrically finite hyperbolic manifold (quotient of $H^n$) has finite Bowen-Margulis measure.
Marc Peign´e, Autour de l’exposant critique d’un groupe kleinien, arXiv:1010.6022v1 constructed examples of geometrically infinite hyperbolic manifolds with finite Bowen-Margulis measure.
He uses a free product/ping-pong construction, and as such the fundamental groups of the manifolds in his exampless have relatively hyperbolic fundamental groups, as of course are fundamental groups of geometrically finite hyperbolic manifolds.
My question: Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and a fundamental group which is not relatively hyperbolic?

ag.algebraic geometry – Rational sections of tropical conics

Let us consider the family of Fermat conics in $(mathbb{C}^*)^2subsetmathbb{C}^2$.
$$picolon V(ax^2+by^2-1)subset(mathbb{C}^*)^2_{a,b}times(mathbb{C}^*)^2_{x,y}to(mathbb{C}^*)^2_{a,b}$$
We know that $pi$ does not admit rational sections: the generic conic is non-split.

Taking tropicalization functor, we get the morpphism $$mathrm{Trop}(pi)colon mathrm{Trop}(ax^2+by^2-1)subsetmathbb{R}^4tomathbb{R}^2$$

Does $mathrm{Trop}(pi)$ admit tropical rational sections? (Sections over $mathbb{R^2}backslash W$ for some proper tropical subvariety $Wsubsetmathbb{R}^2$?)

ag.algebraic geometry – Hodge conjecture for rationally connected/Fano hypersurfaces

We know due to work of Lewis, Murre and others that the rational Hodge conjecture holds for smooth projective hypersurfaces in $mathbb{P}^5$ of degree at most $5$. Does a similar result hold in higher dimension? In particular, is it known if the Hodge conjecture holds for smooth, projective hypersurfaces in $mathbb{P}^{2n+1}$ of degree at most $2n+1$ for some other values of $n$ greater than $2$?