## ag.algebraic geometry – Open immersion of affinoid adic spaces

If $$R$$ and $$S$$ are complete Huber rings with $$varphi: R to S$$ a continuous map, then is it true in general that if $$mathrm{Spa}(S, S^circ) to mathrm{Spa}(R, R^circ)$$ is an open immersion of adic spaces (here $$S^circ$$ and $$R^circ$$ are the power-bounded subrings) then $$mathrm{Spec}(S) to mathrm{Spec}(R)$$ is injective?

For example, this is true if $$R$$ and $$S$$ both have the discrete topology, because if $$frak p$$ and $$frak q$$ are two prime ideals in $$S$$ which are equal after restricting to $$R$$ then $$(frak p, |cdot|_{rm triv})$$ and $$(frak q, |cdot|_{rm triv})$$ (trivial valuations), which are both points in $$mathrm{Spa}(S,S)$$, restrict to the trivial valuation on $$R/varphi^{-1}(frak p)$$.

But I’m not sure how generally to expect that this is true.

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## dg.differential geometry – Vector field along an immersion whose covariant derivative is the differential

Let $$(M,g)$$ be a Riemannnian manifold and let $$f:Sigmato M$$ be a smooth immersion. Then the vector bundle $$f^ast TMtoSigma$$ has a natural bundle metric and metric-compatible connection. Can one characterize the situations in which there must exist a section $$V$$ such that $$nabla V=df$$? This is trivially possible if $$(M,g)$$ is Euclidean space. It feels like it should not be possible in general.

This seems to be equivalent to the existence of a closed 1-form $$omega$$ on $$Sigma$$ and a normal vector field $$w$$ along $$Sigma$$ such that
begin{align}nabla omega-langle h,wrangle&=f^ast g\ h(cdot,omega^sharp)+nabla^perp w&=0end{align}
where $$h$$ is the second fundamental form. I can’t see any immediate conclusions to make.

## role play – Character immersion in Trail of Cthulhu

You say "we usually have a problem … "but it seems from the rest of the question, and forgive me if I'm wrong, that you mean"I am usually faced with a problem … "In other words, have your players really raised the issue as a problem that interferes with their enjoyment of playing?

I have not played or modified the Gumshoe or Cthulhu systems but I know the mechanics and the particular adventure path.

## Out of character

I have to say that a number of things that you raise like OoC or rupture by immersion do not seem to me as such at first glance, maybe there is more than you say, but:

When reading and interpreting these clues, players are largely out of character. I give them an accessory, they read it and discuss it.

It seems to be exactly what the character would do; I certainly cannot see the investigators who find a written clue screwing it up and throwing it away. If the players say "I think …", "And if that means …", "Would you say …" then I can't honestly say if it's the talking player or the character who speaks as the two would do Perfect sense.

the mechanical way refreshes the work to be out of immersion

I don't see anything at all: you have to be in a safe place, time has to pass, or you have to be in between scenarios that all mean you take a break from your activities and that's exactly how people real life refreshes.

You are referring to friends and family. Have you actually played these friends and family with the players in a way that allows them to connect with them? Otherwise, these are just names in a phone book. Human beings do not care about strangers emotionally – that is why humanitarian organizations try to put one face on the disaster relief calls. One person we can bond with is more important than millions we don't know.

## Failure

Failure East an option.

Provided there are enough clues that the character is on a deadline and the players are still not acting with adequate urgency, they fail and unspeakable horrors eat their brains. In fact, this is the most sensible way to end a Cthulhu campaign.

However, don't underestimate how difficult it is to interpret the clues: they seem easy to you because you have the answers. The subtlety is not your friend here: even if you give them a calendar with a red circle around the 12 and a big label saying "END OF THE WORLD DAY!" at least half of the players will miss it, or forget it, or think you are playing with them.

Personally, I think the character should find exactly such a calendar and you should put it in a conspicuous place, crossing every day ostentatiously because they are wasting it. Even then sometimes you I have to break the character and say "Look guys, I really mean it, the world ends on the 12th, you can beat here if you want but you're just wasting time."

As for the reactions of the bad guys: go for it! If players are making choices that will make their progression more difficult, then your role as GM is to allow what the players are doing. If there are now 12 guards on a site instead of 4, let the character find a note saying "The boss says that [the party] is coming: they are at [the place where they hit] there are two days, triple the guard, I want 12 guys on duty at all times! Signed [Underling No 2] ". This tells them that if they were moving faster, things would be easier: it was always up to them to move faster or not.

## Algeic Geometry – Question on Immersion

Let $$X, Y$$ Noetherian full schemes and suppose we have a immersion

$$i: X to Y$$

An immersion is according to https://stacks.math.columbia.edu/tag/01IO a composition of a closed immersion $$c: X to Z$$ by open immersion $$o: Z to Y$$. So in our situation, there is an open subschema $$Z$$ of $$Y$$ such as $$X cong c (X)$$ is a closed subschema of $$Z$$.

My question is if and why the picture $$i (X)$$ is open in $$X: = overline {i (X)} subset Y$$?

The bottom line of my question is that I often read that in the situations described above, if we want to show some topological properties of $$X$$ we reduce the problem to the closed immersion $$X .

## Algebraic Geometry – Possible Closed Immersion of Spec (K)

I'm trying to calculate all possible closed immersions of the scheme $$Spec (K)$$, or $$K$$ is a field? Here is my essay:

assume $$f: X rightarrow Spec (K)$$, is in closed immersion, then $$X$$ must be a point and so $$X = Spec (A)$$, or $$A$$ only has one prime ideal. Since $$f$$ induce a surjective map on structure sheaf which implies that (with closed immersion) that the map induce $$K rightarrow A$$ is surjective, which implies that $$A$$ is $$K$$ module and has dimension as $$K$$ module at most $$1$$. Since the dimension can not be zero, $$A$$ as $$K$$ the module must be isomorphic to $$K$$. So the only closed immersion possible of $$Spec (K)$$ is a map $$Spec (K) rightarrow Spec (K)$$. Am I right?

## dg.differential geometry – Find a surface that is an immersion of \$ Bbb RP ^ 2 \$ in \$ Bbb R ^ 3 \$ based on the algebraic curves

This question is motivated by the Lorenz curve used in economic analysis as well as by the Penrose diagram used in general relativity, used by physicists to visualize cause-and-effect relationships in Minkowski compacted space-time models. .

He is also deeply motivated by Hamiltonian mechanics, symplectic geometry and contact geometry (which can be considered the odd dimensional counterpart of symplectic geometry). Hamiltonian mechanics was a reformulation of Newtonian mechanics. In Hamiltonian mechanics, we study phase spaces of physical systems, symplectic flows, and many other subjects.

From a mathematical point of view, this question is motivated by modern trends in differential geometry, topology, abstract algebra and mathematical physics.

Consider families of flat curves for $$Re (s) ge1:$$

$$zeta: = {(x, y) in Bbb R ^ 2 | x ^ s + y ^ s = 1 }$$
$$tau: = {(x, y) in Bbb R ^ 2 | (1-x) ^ s + y ^ s = 1 }$$
$$psi: = {(x, y) in Bbb R ^ 2 | x ^ s + (1-y) ^ s = 1 }$$
$$phi: = {(x, y) in Bbb R ^ 2 | (1-x) ^ s + (1-y) ^ s = 1 }.$$

Interpretation $$zeta, tau, psi,$$ and $$phi$$ because the phase spaces allow their interpretation as infinite dimensional varieties, in particular of infinite dimensional symplectic manifolds:$$( zeta, omega), ( tau, omega), ( psi, omega), ( phi, omega).$$

The following is a natural Hamiltonian vector field, which defines a Hamiltonian flow on each of the symplectic manifolds.

The process of lifting these varieties into $$Bbb R ^ 3$$ can be realized with homotopic maps. See Paul Frost's answer here: https://math.stackexchange.com/questions/2895816/existence-of-homotopic-map. The essential point to understand is that each of the four symplectic manifolds is a unique projection of the curves recorded by homotopy. In other words, one can project the relief curves on the plane curves in a bijective way. The image provides the intuition of what a single elevator looks like. But in reality, there is an infinity of elevators at different heights above planar curves.

Questions $$($$choose one$$)$$:

$$1)$$ Can we stick together $$zeta, tau, psi, phi$$ including all the elevators and treat it all as a multiple? What would be the complete classification of this variety?

$$2)$$ Can a $$3$$– to be built, whose geodesics are projected on these planar algebraic curves?

I think so, because we can project all the phase-elevated spaces to the corresponding planar phase spaces. I'm pretty sure that the $$3$$-manifold is a submultiple of the infinite dimensional variety.

$$3)$$ Define a surface that is an immersion of $$Bbb RP ^ 2$$ in $$Bbb R ^ 3,$$ which looks like this, and is a three-dimensional analog of the two-dimensional equations above:

This is a form I have constructed to try to get an intuitive understanding of the geometry of variety and phase space in three dimensions:

The Boy's Surface is an example of immersion from $$Bbb RP ^ 2$$ in $$Bbb R ^ 3.$$

The picture concerns the equations because each white strand of the image corresponds to a geodesic strand on a $$2$$– various, namely $$S ^ 2.$$ Each white geodesic also corresponds to a specific plane curve indicated above for a specific value of $$s$$. In the photo $$s = 2.$$ Singularities can be considered as invariants or fixed points because they do not change geographic location for the purposes of this question. variant $$s$$ gauge the appearance of the form. As $$s$$ approach to the infinite form will look like a cube. As $$s$$ tends to $$1,$$ the shape will look like $$2$$ pairs of perpendicular lines located in $$3$$-space.

It may help to say that the image is the shape of the intersections of $$8$$ identical copies of $$S ^ 2$$ with finely many geodesics shown. It's like a venn diagram of higher dimensions if you want.

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