Questions tagged [embeddings]
The embeddings tag has no usage guidance.
179
questions
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45
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Constants in the entropy number of the Sobolev space
For a Sobolev space with $W^s(\Omega)$, where $\Omega\subset R^d$ is a compact space with smooth boundary, we know that the entropy number satisfies $e(\delta, W^s(\Omega, 1),\|\cdot\|_{L_\infty})\leq ...
0
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109
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Embedding of the first Hirzebruch surface in $\mathbb{P}^4$ as a cubic surface
The first Hirzebruch surface (the blow-up of $\mathbb{P}^2$ at one point) is a projective toric surface that naturally embeds into $\mathbb{P}^4$ as a cubic surface (sometimes called the cubic scroll)....
1
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121
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Can a closed null-homotopic curve be filled in by a disc?
Let $U\subseteq\Bbb R^n$ be an open set and $\gamma\subset U$ a closed null-homotopic curve in $U$ (i.e. it can be contracted to a point). Then is there an embedded disc $D\subset U$ with boundary $\...
7
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1
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243
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Does the continuous image of a disc contain an embedded disc?
Let $\phi:\Bbb D^2\to\Bbb R^n$ be a continuous mapping of the 2-disc $\Bbb D^2$ that is injective on the boundary $\partial\Bbb D^2=\Bbb S^1$. Does its image contain an embedded disc with the same ...
1
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82
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Dependence of Sobolev embedding theorem constant on smoothness
Assume that $\Omega \subset \mathbb{R}^d$ is "nice" enough and $k$ is a positive real number. Using the Sobolev embedding theorem, we can get that
$$
\|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
3
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0
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111
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Is it consistent to have an infinite antitone sequence of elementary embeddings such that the involved models include iterated sharps?
$\DeclareMathOperator\crit{crit}$Background essays (the material I've tried to understand in leading up to this question):
Daghighi, et. al. [2014], "The foundation axiom and elementary self-...
0
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1
answer
93
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Does $c$ in the embedding inequality $\|P^\kappa_t \|_{L^p \to L^{p'}} \le c t^{-\frac{d(p'-p)}{2pp'}}$ depend on $\kappa$?
For any $\kappa>0$, we consider the Gaussian heat kernel
$$
p^\kappa_t (x) := (\kappa \pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{\kappa t}},
\quad t>0, x \in {\mathbb R}^d.
$$
Let $L^0 := L^0 (\...
6
votes
1
answer
315
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Different flavours of Vassiliev Conjecture
There is something that puzzles me about "Vassiliev's Conjecture". I am sure I am missing some detail which is obvious to the community, since there are several tightly related kind of ...
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43
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Is a RKHS defined using a feature map over another RKHS bigger than the latter RKHS?
I am interested in learning more about what happens when 'composing' two reproducing kernel Hilbert spaces (RKHS).
Let $\phi\in C(\mathbb{R})$ and $X=[-1,1]^d$. Suppose we have two RKHSs with the ...
4
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0
answers
218
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stating large cardinal axioms in ZF
Can I ask whether there is a good reference for how to state the standard large cardinal axioms in the context of $ZF$? My concern is that it seems that the usual proof that embeddings defined from ...
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0
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118
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Do cycle graphs embed isometrically in spheres?
I recently came across, what seems to be a folklore. Namely, that cycle graphs embeds isometrically into spheres $S^n(r)$, for some $n\in \mathbb{N}_+$ and some $r>0$. However, I could not track ...
3
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0
answers
68
views
Is every weakly $1$-dimensional space embeddable in the plane?
A $1$-dimensional (separable metric) space $X$ is weakly $1$-dimensional if $$\Lambda(X)=\{x\in X:X\text{ is 1-dimensional at }x\}$$
is zero-dimensional (i.e. the space $\Lambda(X)$ has a basis of ...
4
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2
answers
335
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Knot theory in handlebodies of arbitrary genus
It is well known that not all graphs embed on the plane (e.g. the graph $K_{3,3}$). However, every finite graph embeds into a surface of some genus. One can think of this procedure as starting with a ...
6
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144
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Metric spaces containing a topological disc
It is well-known that every connected, locally connected compact metrizable space $X$ contains an arc, that is, a subspace homeomorphic to $[0,1]$. Are there topological properties we can add to these ...
4
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58
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Which cellular embeddings of Eulerian graphs have bipartite duals?
It is well-known that a plane graph $G$ is Eulerian if and only if its (geometric) dual $G^*$ is bipartite.
I am interested in generalisations of this result to cellular embeddings of Eulerian graphs ...
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3
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530
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If a polyhedron in $\mathbb{R}^3$ has local intersections, does it also have more global intersections?
Consider a simplicial complex $K$. A piecewise linear map $f: K \to \mathbb{R}^n$ is an almost-embedding if $f(\sigma) \cap f(\tau) = \emptyset$ for any two disjoint simplices $\sigma,\tau$ in $K$.
...
2
votes
1
answer
206
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Sufficient condition for the union of two submanifolds to be a submanifold
I have two smoothly embedded orientable surfaces $S_1,S_2\subset S^3 \times [0,1]$ with boundary such that
$(i)$ $S_1\cap S_2$ is a smoothly embedded surface without boundary and
$(ii)$ $\overline{...
4
votes
0
answers
138
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Applications of Strong Whitney Embedding
I am looking for applications of the strong Whitney's embedding theorem that have an advantage over weak theorems. That is, applications where it's important that the dimension of the Euclidian space ...
4
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259
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Does this "join-like complex" of $K_5$ and $K_3$ embed in $\Bbb R^4$?
Consider the following 2-dimensional CW-complex: its 1-skeleton is $K_8$, which we write as an edge-disjoint union $K_5\cup K_{5,3}\cup K_3$.
Then for any two edges $ab\in E(K_5)$ and $cd\in E(K_3)$ ...
2
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0
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114
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How to express Kunen's inconsistency, Reinhardt and Wholeness axioms, by single sentences?
Working in $\sf NBG, $ can we express the property of a class being set theoretically definable, by a single sentence? Like for example, the following way:
$$\operatorname {std}(X) \iff \exists x_1 \...
-3
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1
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66
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Sobolev embedding [closed]
I was trying to understand Sobolev embedding, some results about this topic are not clear to me.
My question is the following:
what are the condition on $p_1 , \alpha_1, p_2 $and $\alpha_2$ for
$W^{...
4
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0
answers
242
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Can we have full choice prior to Reinhardt cardinals?
Working in $\sf ZF + Reinhardt \ cardinal$, can we have full choice over all stages $V_{\alpha < \kappa}$ where $\kappa$ is the Reinhardt cardinal, i.e., the critical point of the elementary ...
1
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65
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Can this method let choiceless large cardinals be smaller than cardinals compatible with choice?
Recall question "Can we have this sequence where choice fails and returns?"
Can that theory be extended with requiring the $\mathcal V_n$'s to fulfill a choiceless large cardinal extension ...
2
votes
1
answer
329
views
Can we interpret Reinhardt cardinals this way?
To the language of set theory add a primitive unary predicate $\operatorname {Universe}$ and a primitive total unary function $j$. Add all axioms of $\sf ZF$ in the language of this theory, i.e. the ...
1
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2
answers
258
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Can we iteratively reflect on self elementary embeddable stages of the cumulative hierarchy?
Is it possible to iterate elementary embeddability and reflect on those stages that are elementary embeddable to themselves?
The following is a formal capture of that idea:
To the language of $\sf ZF$...
8
votes
1
answer
1k
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Is there a form of choice that can elude Kunen's inconsistency theorem?
When it is said that Kunen inconsistency theorem proves that given $\sf ZFC$ no elementary embedding can exist from the universe to itself. Most references quote full choice in stating that result, ...
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104
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Name for homotopy totalization of Goodwillie tower (in embedding calculus)
Let $M,N$ be a manifold and consider the presheaf of spaces $\textrm{Emb}(-, N)$ on the open sets of $M$. Classical embedding calculus produces a goodwillie tower
$$ \ldots \rightarrow T_{k+1} \textrm{...
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1
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86
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Hadamard submanifolds of $k$-fold product of hyperbolic plane
Let $\kappa>0$ and $d,k$ be a positive integers with $k\ge d$. For $k\in \mathbb{Z}^+$ large enough, can one find a geodesically complete and simply connected $d$-dimensional Riemannian ...
6
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145
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What is the minimum $n$ for which $\Bbb H^3$ can be isometrically embedded in $\Bbb R^n$ as a bounded set?
Consider the hyperbolic $3$-space $\Bbb H^3$ (i.e., the unique, simply-connected, $3$-dimensional complete Riemannian manifold with a constant negative sectional curvature equal to $-1$). The Nash ...
3
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1
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268
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If we add stratified\acyclic replacement to the wholeness axiom, would that increase its consistency strength?
If we add to the wholeness axiom, the axiom of stratified\acyclic replacement, what would be the consistency state and strength of the resulting theory?
The wholeness axiom $\sf WA$, introduced by ...
7
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1
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322
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Can we have mutual elementary embeddability between distinct transitive sets?
Is it consistent with $\sf ZFC$ to have mutual elementary embeddability between distinct transitive sets?
Formally, is the following theory consistent? $${\sf ZFC} + \exists M \exists N: M,N \text { ...
5
votes
1
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193
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How to formalize this isotopy?
I'm studying the H-Cobordism theorem following the Lectures of John Milnor, and in the proof of the Whitney trick for cancel pairs of self-intersection points I have the next problem with an isotopy ...
4
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97
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Mostow-Palais equivariant embedding for manifolds with corners
Let $M$ be a compact smooth manifold and let $G$ be a connected compact Lie group acting on $M$. According to an old theorem of Mostow and Palais, there exists a $G$-equivariant embedding of $M$ into ...
4
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0
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246
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Explicit formula for embedding of real projective spaces into Euclidean spaces
I am interested in representing vectors in $\mathbb{R}^n$ in a sign-invariant and efficient manner. That is, I am looking for a function
$$f:\mathbb{R}^{n+1}\rightarrow\mathbb{R}^d$$
such that for $v\...
5
votes
1
answer
228
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Space of spacelike embeddings as infinite-dimensional manifold
Consider a four-dimensional Lorentzian manifold $(\mathcal{M},g)$ and a $3$-dimensional compact manifold $\Sigma$, such that there exists a spacelike embedding $i:\Sigma\to\mathcal{M}$ so that $h:=i^{\...
3
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1
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207
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Adding a self-intersection point to an immersion
I'm currently working on strong Whitney's embedding theorem, using Adachi's notes on Embeddings and Immersions. But I am stuck on a statement that Adachi comments without proof about adding a unique ...
1
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0
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115
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Embedding (Kronecker product) preserves the structure?
In PGL$_{n}(\textbf{C})$, conjugacy classes of toral involutions can be represented by $$s_{i} = \begin{pmatrix}
-I_{i} & 0\\
0 & I_{n-i}
\end{pmatrix}.$$ for $1 \leqslant i \leqslant [n/2]$. ...
8
votes
1
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372
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Whitney embedding theorem for Hölder manifolds
According to a result of Whitney any $C^r$-manifold, $r\geq 1$, is $C^r$-homeomorphic to a smooth embedded submanifold of some Euclidean space; see Theorem 1 in
Hassler Whitney, "Differentiable ...
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101
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Exists $G$-equivariant embedding with faithful representation of $G$?
Let $k$ be a field of characteristic zero and $G$ a reductive group over $k$. Furthermore, let $X$ be a projective $k$-variety with a $G$-action. Then we know, for example from Mumfords book about GIT,...
5
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840
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Can a smooth manifold be realised as the image of a smooth function?
Consider, $M$, a smooth $m$ dimensional submanifold of $\mathbf R^n$. Does there exist a smooth map $X: \mathbf{R}^m\to\mathbf R^n$ such that $M=X(\mathbf R^m)$?
$X$ may have points at which the ...
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2
answers
217
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Ricci scalar of submanifold of $\mathbf R^n$
Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations:
\begin{equation}
f_1(\vec x)=0, \\
\vdots \\
f_m(\vec x)=0,
\end{equation}
where $\vec x$ are ...
2
votes
1
answer
268
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Volume of submanifold as integral of delta-function
Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations:
\begin{equation}
f_1(\vec x)=0, \\
\vdots \\
f_m(\vec x)=0,
\end{equation}
(where $\vec x$ are ...
3
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0
answers
177
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Volume of sub-manifold of $\mathbf R^n$
Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of polynomial equations:
\begin{equation}
P_1(\vec x)=0, \\
\vdots \\
P_m(\vec x)=0,
\end{equation}
(where $\...
3
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1
answer
179
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Topologically embed Klein bottle into $\mathbb{R}^4$ projecting to usual "beer-bottle" surface in $\mathbb{R}^3$
(Originally asked in 2018 at https://math.stackexchange.com/questions/2946505/topological-embedding-of-klein-bottle-into-mathbbr4-that-projects-to-usual?noredirect=1#comment9514257_2946505;cross-...
4
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207
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Does a smooth relative curve $X/S$ embed into $\mathbb{P}^3_S$?
Suppose that $\pi:X\to S$ is a smooth projective morphism of relative dimension 1. If $S$ is the spectrum of an algebraically closed field, then it is known that $X$ embeds into $\mathbb{P}^3_S$.
...
3
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152
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Constant in Naor and Neiman's Assouad Theorem
In Naor and Neiman's Assouad embedding theorem - "Assouad’s theorem with dimension independent of the snowflaking" Revisita Mathematica, the authors derive quantitative estimates on the ...
3
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0
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87
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Is the thickening of a PL 2-disc in $\Bbb R^4$ a 4-ball?
Let $D\subset\Bbb R^4$ be a PL-embedded 2-dimensional disc. Let $N=D+K$ be a thickening of the disc, where $K$ is some sufficiently small 4-dimensional PL-ball and "$+$" means Minkowski ...
1
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0
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135
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Realization of a subgroup in a maximal subgroup of a classical group
$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}$In the finite group $G = \operatorname{PGL}_8(5)$, there is an elementary abelian $2$-subgroup $E$ of rank 5. $E = A_{1} \times A_{2} $ where $...
4
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answers
166
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In how far does the Whitney trick work in the piecewise linear setting in $\Bbb R^4$?
I usually read about the Whitney trick in the context of smooth manifolds, but I wonder in how far it works in the piecewise linear (PL) category as well. I have a specific setting in mind that I will ...
3
votes
1
answer
116
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Holomorphic/Symplectic embedding of Riemann surfaces
Let $\Sigma_g$ denote a Riemann surface and let $X$ denote the complex surface $\Sigma_g \times \Sigma_g$. Then can there exist holomorphic embeddings of $\Sigma_l$ into $X$ for $l < g$?
What about ...