Is this $\mathbb P^1$-fibration over $\mathbb P^1$ a flat family?

This example is a test of several quantities invariant in a flat family.

Let $ X$ be a projective variety, let $ D$ be a divisor on $ X$ with $ \dim|D|\ge 1$ . Choose a pencil $ \{X_t\}_{t\in\mathbb P^1}$ inside $ |D|$ . This is in particular an algebraical family, therefore a flat family. We know Hilbert polynomial is invariant in a flat family, therefore all quantities arising from Hilbert polynomial, including dimension, degree and arithmetic genus, should be invariant in a such a family.

Here is an example seems contradicting to me: Let $ p:X\to \mathbb P^1\times \mathbb P^1$ be the surface blowing up at a point, and let $ \pi:X\to \mathbb P^1$ be the composite $ p_1\circ p$ with $ p_1$ the projection to the first coordinate. ($ X$ is indeed a Hirzebruch surface $ \Sigma_1$ , but we are considering its $ \mathbb P^1$ -fibration in a different way.)

Then the map $ \pi$ gives us a pencil family of divisors on $ X$ , with general fiber $ X_t=\mathbb P^1$ , and special fiber $ X_0=\mathbb P^1\cup \mathbb P^1$ , two $ \mathbb P^1$ intersecting at a point. Now, let’s check the those quantities mentioned above.

Obviously, the dimension is always 1 which doesn’t change. The arithmetic genus is also the same by the answer of this post. However, the degree of $ X_t$ is $ 1$ , while the degree of $ X_0$ is $ 2$ (The degree is additive on irreducible components, I believe.)

So my question is:

(1) Does $ \pi: X\to \mathbb P^1$ gives a flat family of divisors?

(2) Is it true that $ \deg(X_0)=2$ ?

Flat or linkless embeddings of graph with fixed projection

The problem of finding whether a given planar diagram of a graph, with over- and under-crossings, is a linkless embedding or not has unknown complexity (Kawarabayashi et al., 2010). My first question is, what about finding whether a given diagram is a flat embedding?

There is also interesting work showing how links and knots are “inevitable” in given graph projections, in the sense that any assignment of over- or under-crossings yields a knotted or linked graph embedding (Taniyama and Tatsuya, 1996). There’s a wonderful graph-minor characterization of whether a given graph, with no fixed projection, has a linkless or flat embedding (Robertson et al., 1995). My second question is: have people attempted to design algorithms (even for specific cases or with bad complexity) to decide whether a given projection (without over- and under-crossings) can be given crossing information to yield a linkless or flat embedding?

Such an algorithm would seem hard to design at first, because it’s already hard to recognize if a given diagram is linkless, but that doesn’t seem to rule out an algorithm out for special classes of graphs because there may be “easy” linkless or flat embeddings to attain in those cases.

References

(Kawarabayashi et al., 2010) Kawarabayashi, Ken-ichi; Kreutzer, Stephan; Mohar, Bojan, Linkless and flat embeddings in 3-space and the unknot problem, Proceedings of the 26th annual symposium on computational geometry, SoCG 2010, Snowbird, UT, USA, June 13–16, 2010. New York, NY: Association for Computing Machinery (ACM) (ISBN 978-1-4503-0016-2). 97-106 (2010). ZBL1284.05174.

(Robertson et al., 1995) Robertson, Neil; Seymour, Paul; Thomas, Robin, Sachs’ linkless embedding conjecture, J. Comb. Theory, Ser. B 64, No. 2, 185-227 (1995). ZBL0832.05032.

(Taniyama and Tatsuya, 1996) Taniyama, Kouki; Tsukamoto, Tatsuya, Knot-inevitable projections of planar graphs, J. Knot Theory Ramifications 5, No. 6, 877-883 (1996). ZBL0876.57011.

Moduli space of flat connections of Lie group over a 2-torus

We know that the moduli space of SU($ N$ ) flat connections over a torus, is equivalent to a complex projective space $ \mathbb{P}^{N-1}$ Namely, $ $ M_{\rm flat}=\mathbb{E} / {S}_N = \mathbb{P}^{N-1} $ $ where $ \mathbb{E}$ is given by $ $ \mathbb E := \left\{ (\phi_1,\cdots, \phi_N) \equiv {(\mathbb T^2)}^N; \text{ subject to a constrain }\sum_i \phi_i=0 \right\} . $ $ while the $ {S}_N$ is the symmetric group usually denoted as $ S_N$ (the Weyl group of SU$ (N)$ ).

I learned the answer from the post and Lisa Jeffrey’s note: Moduli space of flat connections over a Riemann surface

My questions

  • what is the moduli space of SO(N) flat connections over a 2-torus?

  • what is the moduli space of PSU(N) flat connections over a 2-torus?

If this is too general, we can focus on the case: SO(3)=PSU(2).

Thank you for the kind comments and helps!


Some Refs I found:

The moduli space of flat SU (2) and SO (3) connections over surfaces

https://faculty.math.illinois.edu/~michiel2/docs/thesis.pdf

Flat spherical orbifolds

What is known about existence and classification of flat spherical orbifolds?. Here I mean orbifolds that admit a flat Riemannian metric (Euclidean orbifolds) and whose underlying topological space (their naive quotient space) is homeomorphic to an $ n$ -sphere.

In dimension $ n = 2$ there is a classical classification (recalled e.g within table 1 of arXiv:1705.08431) listing, up to scale and isomorphism, the pillowcase orbifold and three more.

In higher dimensions it seems that potentially relevant results mostly come from branched covering theory. If we can realize the $ n$ -torus as a branched cover over the $ n$ -sphere such that the action of the group of deck transformations is smooth, then the corresponding homotopy quotient of the $ n$ -torus by the group of deck transformations should be a flat spherical orbifold. And for the action of the group of deck transformations to be smooth, it should be sufficient that the branching locus is smooth. I suppose.

For $ n = 4$ there is a result in this direction in

  • Massimiliano Iori, Riccardo Piergallini, 4-manifolds as covers of the 4-sphere branched over non-singular surfaces, Geom. Topol. 6 (2002) 393-401 (arXiv:math/0203087)

I keep feeling a bit unsure about some technical fine print in the definitions there, but that their result implies the desired statement of existence of a smooth branched covering with smooth branching locus of, in particular, the 4-torus over the 4-sphere, is asserted in

  • Ali Chamseddine, Alain Connes, Viatcheslav Mukhanov, Geometry and the Quantum: Basics, JHEP 12 (2014) 098 (arXiv:1411.0977)

  • Alain Connes, Geometry and the Quantum, Foundations of Mathematics and Physics One Century After Hilbert. Springer 2018. 159-196 (arXiv:1703.02470)

on p. 24 and p. 30, respectively (see also footnote 5 in the former).

So it looks like we may conclude that 4-dimensional flat spherical orbifolds exist. (Maybe that’s actually wrong, due to fine print I am overlooking, or it’s actually trivial, due to general facts that I am missing. If anyone has more, please let me know.) If that is right, can we say anything about the space of available choices?

And how about other dimensions? (I’d be particularly interested in dimensions $ \leq 10$ .)

Flat Rate Cargo in Book 7 Trade System

Classic Traveller Book 7: Merchant Prince describes on pages 34 through 42 a trade system focused on speculative cargoes. There are, however, a few references to carrying, rather than purchasing and reselling, cargo at a flat rate of 1,000 Cr per ton. Specifically the “Ship Revenue” and “Cargo” tables on page 39 appear to relate to this. I am unable to find any text related to the “Cargo” table, specifically, text describing what constitutes “major,” “minor,” and “incidental” cargoes.

What do these cargo categories represent, and if tonnage as one might infer, what tonnages do they equate to?

I’m comfortable inventing these for my game, but here I’m asking what has been written by the designers on this that I’m unable to find.

Finite generation of flat deformations of algebras

Let $ R=\mathbb C[q^{\pm 1}]$ and let $ A$ be a graded (possibly non-commutative) $ R$ -algebra, $ A=\oplus_{n=0}^\infty A_n,$ where $ A_0=R$ and all $ A_n$ ‘s are free $ R$ -modules. Then $ A’=A/(q-1)$ is a graded algebra over $ R/(q-1)=\mathbb C$ and so $ A$ can be thought as a deformation of $ A’.$ I couldn’t find a definition of a flat deformation of algebras but I would imagine this being an example of it, correct? (Assume that $ A’$ is commutative if it helps.)

Main question: Is it possible that $ A’$ is finitely generated while $ A$ is not (as algebras over $ \mathbb C$ and $ R$ respectively)?

Flat norm metrizes the weak* topology

I’ve come across the following statement in literature (without proof or reference) about the flat norm of currents $ $ F(T) = \sup \{ T(\omega) : \omega \in D^k(U), |\omega(x)| \leq 1, |d\omega(x)| \leq 1 \}: $ $

The importance of the flat norm is due the fact that (at least in the space of normal currents with a bound on the mass of the current and on the mass of the boundary) it metrizes the weak* topology.

Is there a reference for this? If not, I would be happy about hints how one would one go about showing this. I have been looking into proofs which show that the Wasserstein-1 distance metrizes the weak*-topology of probability measures but they seem difficult to adapt to that case.

Convert flat object hierarchy to json

I do work for a project currently, where the data is send to the server as application/x-www-form-urlencoded (which is bad, and it should be JSON, but unfortunately I am not able to change this one).

The following code snipped will transfer the given parameters (in a loop) to a Map, which can then be transformed to json (eg. by using jackson).

I would like to make this even better, so please post your comments and suggestions.

Parameter-List:

_id=[5bfad95450642c333010daca],  _rev=[1-9ce33949c3acd85cea6c58467e6a8144],  type=[Group],  user=[aUSer],  default=[aDetail],  store[aDetail][prop]=[5],  store[aDetail][lprop1][0][time]=[00:00],  store[aDetail][lprop1][0][value]=[14],  store[aDetail][lprop1][0][timeAsSeconds]=[0], store[aDetail][lprop1][1][time]=[07:00],  store[aDetail][lprop1][1][value]=[8],  store[aDetail][lprop1][1][timeAsSeconds]=[25200],  store[aDetail][anprop]=[25],  store[aDetail][lprop2][0][time]=[00:00],  store[aDetail][lprop2][0][value]=[61],  store[aDetail][lprop2][0][timeAsSeconds]=[0],    store[bDetail][prop]=[6],  store[bDetail][lprop1][0][time]=[00:10],  store[bDetail][lprop1][0][value]=[12],  store[bDetail][lprop1][0][timeAsSeconds]=[0], store[bDetail][lprop1][1][time]=[07:10],  store[bDetail][lprop1][1][value]=[9],  store[bDetail][lprop1][1][timeAsSeconds]=[25200],  store[bDetail][anprop]=[25],  store[bDetail][lprop2][0][time]=[00:00],  store[bDetail][lprop2][0][value]=[61],  store[bDetail][lprop2][0][timeAsSeconds]=[0],  created_at=[2018-01-11T20:48:22.574+0100], ... 

Code-Snippet:

fun parseToMap(map: MutableMap<String, Any>, key: String, value: Any): MutableMap<String, Any> {     val cleanedV = if (value is String) URLDecoder.decode(value, "UTF-8") else value      if (!key.contains("[")) {         map.putIfAbsent(key, cleanedV)     } else {         // mapKey is the key going to get stored in the map         val mapKey = key.substring(0, key.indexOf("["))          // nextKey is the next key pushed to the next call of parseToMap         var nextKey = key.removePrefix(mapKey)         nextKey = nextKey.replaceFirst("[", "").replaceFirst("]", "")          var isArray = false         var index = -1          if (nextKey.contains("[") &&                 nextKey.substring(0, nextKey.indexOf("[")).matches(Regex("[0-9]+"))) {             index = nextKey.substring(0, nextKey.indexOf("[")).toInt()             isArray = true         }          // mapkey used for object in list         val newMapKey = if (isArray) nextKey.substring(nextKey.indexOf("[") + 1, nextKey.indexOf("]")) else ""          val child: Any?         var childMap: MutableMap<String, Any> = mutableMapOf()          if (map.containsKey(mapKey)) {             println("key $  mapKey exists already")             child = map[mapKey]              when (child) {                 is MutableList<*> -> {                     if (child == null || child.isEmpty()) {                         childMap = mutableMapOf()                         val tmpList = child as MutableList<Any>                         tmpList.add(childMap)                         map.put(newMapKey, tmpList)                     } else {                         if (child.size > index) {                             childMap = child.get(index) as MutableMap<String, Any>                             childMap = parseToMap(childMap, newMapKey, value)                         } else {                             childMap = parseToMap(childMap, newMapKey, value)                             val tmpList = child as MutableList<Any>                             tmpList.add(childMap)                         }                     }                 }                 is MutableMap<*, *> -> childMap = map.get(mapKey) as MutableMap<String, Any>             }         } else {             if (isArray) {                 child = mutableListOf<Any>()                 childMap = parseToMap(childMap, newMapKey, value)                 child.add(childMap)                 map.put(mapKey, child)             } else {                 childMap = mutableMapOf<String, Any>()             }         }          if (!isArray) parseToMap(childMap, nextKey, value)          map.putIfAbsent(mapKey, childMap)     }      return map } 

Calling this method:

    decodedParameters.forEach { k, v -> run {         val cleanedV = v.toString().replace("[", "").replace("]", "")         jsonMap = parseToMap(jsonMap, k, cleanedV)     } } 

Simplest complete combinator basis pair for flat expressions

In Chris Okasaki’s paper “Flattening Combinators: Surviving Without Parentheses” he shows that two combinators are both sufficient and necessary as a basis to encode Turing-complete expressions without the need for an application operator or parentheses.

Compared to John Trump’s encodings of combinatory logic in “Binary Lambda Calculus and Combinatory Logic” through prefix coding S and K combinators with an application operator, only needing two combinators for flat expressions increases the code density to optimality. The resulting Goedel numbering maps every integer to a valid, well-formed closed-term expression unlike most calculi and minimal description length relevant esolangs whose canonical representations usually permit descriptions of syntactically invalid programs.

However Okasaki’s encoding was meant to be most helpful in a one-way mapping from lambda calculus terms to bitstrings, not necessarily the other way around as the two combinators used in this reduction are relatively complex when used as practical substitution instructions.

What is the simplest complete combinator basis pair that does not require an application operator?