On the singular loci of higher secant varieties of Veronese embeddings
-
Katsuhisa Furukawa
Abstract
The 𝑘-th secant variety of a projective variety
1 Introduction
Throughout the paper, we work over ℂ, the field of complex numbers.
Let
where
The construction of secant varieties (or more generally, join construction of subvarieties) is not only one of the most famous methods in classical algebraic geometry, but also a very popular subject in recent years, especially in connection with fields of research such as tensor rank and decomposition, algebraic statistics, data science, geometric complexity theory, and so on (see [22, 23] for more details).
Despite of a rather long history and the popularity, most of the fundamental questions on the higher secant varieties
In this paper, we concentrate on the case of
The knowledge on singularities of higher secant varieties is fundamental and very important for its own sake in the study of algebraic geometry and also can be useful for problems in applications. For example, it can be used as a key condition to establish the identifiability of structured tensors (e.g. the introduction in [7] and references therein).
For any irreducible variety
unless
There are some known results on the singular loci of 𝑘-th secant varieties
For the case of Veronese varieties, it is classical that
In the present paper, we explore the singular locus of any higher secant variety of the Veronese variety, and introduce a new main origin for the singularity other than the trivial singularity. We call this the “subsecant locus”. As in our main results, these loci show an interesting trichotomy phenomenon among generic smoothness, non-trivial singularity, and trivial singularity.
For any given point
It naturally forms an increasing sequence of loci in the 𝑘-th secant variety as
In particular, we have that
Thus a basic question for our concern could be stated as follows: for given
In principle, it is somewhat straightforward (despite the computational complexity) to check the singularity, once a complete set of equations for a higher secant variety is attained. But, as mentioned above, not much is known about the defining equations and they seem quite far from being fully understood at this moment, even for the Veronese case (see [24, 11] for the state of the art). Due to the lack of knowledge on the equations for the higher secant variety, it is very difficult to determine the singular locus in general.
In this paper, without further understanding on the equations (!), we introduce a geometric way to pursue it for this kind of problems, which is based on a careful study on the behavior of embedded tangent spaces moving along a locus in the Veronese variety. For the case
Let
If
If
If
Concerning singular points of arbitrary
Let
For
we have the following.
If
If
If
In the same situation as Theorem 2, for
if 𝑘 is in one of the ranges named (i), (ii), (iii) in Table 1, then the following property corresponding to the name of the range holds.
(Non-)singularity of
|
|
|
(i) | (ii) | (iii) |
|---|---|---|---|---|
|
|
5 |
|
None |
|
|
|
7 |
|
|
|
|
|
28/3 |
|
|
|
|
|
5 |
|
|
|
|
|
35/4 |
|
|
|
|
|
14 |
|
|
|
|
|
7 |
|
None |
|
|
|
28/3 |
|
|
|
To understand the reason for considering the conditions that
Set
By [2], the codimension of
We say a point
We make some remarks on the theorems above.
(a) In the case of
(b) Theorem 1 (i) is stronger than Theorems 2 (i) and 3 (i), since it claims smoothness for “every point” in the 𝑚-subsecant locus
(c) Theorems 1 (i), 2 (i), and 3 (i) correspond to the 𝑘-identifiable case of a general point
(d)
As an application of our main results for
Theorem 5 (Singular locus for
σ
4
(
v
d
(
P
n
)
)
)
Let
If
For
For a projective variety
Example 6 (Cases with a nice description)
The smallest case for the singular locus of
For the case
Such a simple description of the singular locus can be attained in a few more cases (see Corollary 32 for details).
The paper is structured as follows.
In Section 2, as preparation, we first recall some preliminaries on 𝑘-th secant varieties and corresponding incidences.
Then, using projective techniques, such as Terracini’s lemma, the trisecant lemma, descriptions of embedding tangent spaces, and tangential projections, we reveal several geometric properties of 𝑚-subsecant varieties in higher secant varieties of Veronese varieties, which are crucial for the proof of the main theorems.
In Section 3, as an illustration of the whole picture and our main ideas, we treat the case
2 Some geometric properties of subsecant varieties
2.1 Projection from the incidence to the secant variety
For a (reduced and irreducible) variety 𝑋, we denote
For the 𝑑-uple Veronese embedding
where we write
We also have
We have some remarks on the incidence variety
(a)
(b) In the case of
we have
(c) (Alternative incidences for the 𝑘-th secant variety)
In the literature, instead of
Let us fix an 𝑚-plane
(a) Let
then we set
In particular, when
(b) On the affine open subset
Let us study the behavior of some points in the boundary of 𝐼, which belong to 𝐼 but do not belong to
Let
For
there is a subset
Proof
Let
Let
where
we may regard
Under
We consider the projection
where 𝑞 is equal to
If an irreducible component
Now, let
If
is determined in
is determined and is contained in
Assume
coincide since it holds on an open subset of 𝑆.
Let
be the fiber of
we have
Lemma 10 (Non-triviality of subsecant varieties)
For an 𝑚-plane
In particular,
Proof
Let
Since
Take
We have some consequences of Lemma 10.
(a) (Border rank preserving pair)
For any
as a set.
Since one inclusion is obvious, let us prove
In other words, for any 𝑑-th Veronese embedding
(b) (Every
Let
Therefore, we obtain that
2.2 General secant fiber of a subsecant variety, entry locus, and its Veronese image
We take another incidence variety
with the projections
where
For
Let
We begin with a consequence of Terracini’s lemma in our setting and add two more lemmas concerning “the entry locus”
Assume that
Proof
Since
For a projective variety 𝑍, let
Proof
For simplicity, we set
Let
Let
Since
is generically finite.
Let
is dominant (this is because, for general
Let 𝐹 be an irreducible component of
For general
The 𝑘-tuple
which contradicts that 𝜌, given in (2.7), is generally finite. ∎
For a projective variety
Proof
Since
Now let us focus on the case
the image of the 𝑑-uple Veronese embedding of
Let
Assume
If
for the preimage
If
(a) Two inequalities
are equivalent to
this occurs if and only if
(b) In Proposition 15 (ii), if
is
To prove Proposition 15, we settle two lemmas, Lemmas 17 and 19; the former one is technical and the latter geometric.
Let
If
If
If
Note that Lemma 17 (iii) is applied in a discussion of the proof of Theorem 2 (ii), though it is not used in this section.
To show the lemma, we need some calculations as follows.
(a) Let
and then
Then the possible values of
modulo 3, which is absurd.
(b) For
For
Proof of Lemma 17
(i) From Remark 18 (a), we may assume
Since
using
Setting
Then
On the other hand, when
Hence
if
(ii) Let
and
If
this case does not occur since 𝑘 cannot be an integer.
If
In this case,
(iii) Next, we assume
Using
If
we also have
The next lemma concerns a general fact on linear sections of Veronese varieties, which is of independent interest itself.
Let
be the 𝑒-uple Veronese embedding of
Assume
(2.9)Assume
where the left-hand side means dimension of component(s) passing through
Assume
Proof
Let
be the linear projection from the (
In particular,
For the 𝑘-plane
where, by generic smoothness, the right-hand side is equal to
The condition
from the 𝑚-plane
By Terracini’s lemma, for general
is of dimension
Let
again the trisecant lemma implies that the right-hand side in (2.10) is only the set of
for some
Finally, we consider the case of
satisfies the commutative diagram
where
If
hence the trisecant lemma implies
In the diagram above, for any
indeed,
Since
We give one calculation before proving Proposition 15.
For
(as under the conditions of Proposition 15 and Remark 16 (a)), then
If
Proof of Proposition 15
(i) For simplicity, we set
For
From Lemma 13, we have
Let
Let
is contained in the linear variety
and is of codimension
In particular,
Since
Assume
we have
In particular, we have
Next, let us consider the case of
is a linear subvariety of dimension at most 𝑘.
Since
is of dimension at most 1.
On the other hand, since
as in Remark 20. For the union
we see that
Let
that is to say, there is a curve
then
then
Hence
(ii) In the case when
we have
We end this subsection by making one more important remark on the case when
is secant defective, which will be used in the proof of Theorem 3 (ii).
Remark 21 (Estimate in defective cases)
For four defective cases
similarly to Proposition 15, we can have an estimation
where
For three cases
we see that
by the trisecant lemma so that we may take
By Lemma 19 (ii), we get
where
Thus, by the intersection argument in
For the remaining case
as follows.
For the 6-dimensional subspace
the 0-dimensional intersection.
Then
and set
2.3 Estimate for the linear span of tangents moving along a subsecant variety
First, we give the following explicit description of the embedded tangent space
Recall that
where
and “∗” means the remaining monomials.
Let
using (2.13), where
In particular, in case of
As a consequence, we settle a key proposition which estimates a lower bound of the dimension of the linear span of moving embedded tangent spaces along a subset of a given
Let
is greater than or equal to
where
Proof
For a given
as subsets in
Since
is of dimension
Again, by (2.15), we see that
3 Case of
m
=
1
3.1 Symmetric flattening and conormal space computation
For the proof of Theorem 1, we begin with some preliminaries on equations for secant varieties and conormal space computation via known sets of equations, whereas we mainly adopt the geometric viewpoint and techniques for the
Let 𝑉 be an
Given a form 𝑓 of degree 𝑑, the minimum number of linear forms
Now, we recall that some part of defining equations for
with
for any 𝑎 with
in the bases defined above.
We call this “the
It is obvious that if 𝑓 has (border) rank 1, then any symmetric flattening
Let us recall some more basic terms and facts.
Let
and the equality holds if and only if 𝑍 is smooth at 𝑝.
This conormal space is quite useful to study the (embedded) tangent space
For any given form
It is straightforward to see that
Suppose that
Then, for any 𝑎 with
as a subspace of
Proof
Let
it is well known that, for any
in
Further, since
where
The assertion is immediate when we apply this fact to a partial polarization
because
3.2 Proof of Theorem 1
Now we study singularity and non-singularity of the subsecant variety
Proof of Theorem 1
(i) Let 𝑓 be any form belonging to
We claim that
Now, let us show that
where the left-hand side is given by the expected codimension of the 𝑘-th secant variety. By Proposition 23, we also have
Thus 𝑓 is a smooth point of
Since
(a) If 𝑑 is odd and
where
So we obtain
which tells us that
(b) When 𝑑 is odd and
Thus, by a dimension counting similar to case (a), we see that
which coincides with the expected codimension as desired.
Thus 𝑓 is a smooth point of
(ii) First note that
Now, let
Then a general point
is a smooth point of
Taking
which is equivalent to the formula
Finally, since
(note that
(iii) By assumption,
hence the assertion follows.
(iv) For
Part (iv) is the exception to the trichotomy in Theorem 1.
Under the condition
4 Proof of main results
In this section, we prove Theorems 2 and 3. We will first discuss the non-singularity result and then the results for the singular loci.
4.1 Generic smoothness
We begin with a lemma which deals with a secant fiber of a general point in an 𝑚-subsecant variety
Assume
(i.e.,
Proof
(i) Consider any
regarding 𝑎 and
As in Remark 8, set
For the affine open subset
where for
Since
which gives
(ii) Let
Note that, by the linear independence of
if and only if
Let
We may assume
Since
We take
for each 𝜉, and consider the ideal
with
where
Using the discussion of (i), we have
Therefore,
We recall some known results on the 𝑘-the secant variety and its incidence in terms of 𝑘-fold symmetric product of
It is known that
Assume
Now, we are ready to prove Theorem 2 (i) and Theorem 3 (i).
Proof of Theorem 2 (i) and Theorem 3 (i)
For an 𝑚-plane
Thus, by the assumptions of Theorems 2 and 3, we know that
that is,
In particular, under the assumption
If
Let
Now, we restrict the projective morphism
Since
which implies
which implies that 𝑎 is a smooth point in 𝑌. ∎
We present an example which shows that one cannot extend this generic smoothness result to an arbitrary point in the locus
Example 27 (Singularity can occur at a special point in Theorem 3 (i))
Let
and let
Note that
Note that, using parameterization (2.13), we can estimate the dimension of the right-hand side of (4.2).
Take an affine open subset
On
and at
which shows that
Thus, by (4.2), we obtain
4.2 Singularity
In this subsection, we will prove parts (ii) and (iii) both in Theorem 2 and Theorem 3, which show the singularity of the 𝑚-subsecant loci
Proof of Theorem 2 (ii) and Theorem 3 (ii)
As we noted above, it is enough here to show that
We will first prove that
next for Theorem 3 with
or
can be directly obtained at the end by Lemma 10.
Take a general point
Suppose
Terracini’s lemma implies
First of all, let us consider Theorem 2 (ii) with
Set
Then
as in Remark 16.
We have
For case (a1) (i.e.,
From inclusion (4.3), we obtain
which implies
Now, assume
we have
For
For
The condition
Secondly, let us regard Theorem 3 (ii).
For
For case (b1), i.e., the defective case, it is known that all the
where 𝛽 is equal to
which contradicts
For case (b2), i.e., just after the defective case (b1), the 𝑘-th secant variety
which fails to hold in (b2); more precisely, for
the value
respectively, which must be greater than 0 because of the condition
Now, we discuss the following two cases:
Similarly, suppose that
and take a general point
(modulo permutation on
Setting
For the
Then we have
where the right-hand side must be
Again by (4.6) and Proposition 22, we get
which is a contradiction since
Note that, for
i.e., just after the non-identifiable case (c2), the singularity is already shown in the second part of this proof, where
Finally, since
We finish this section by proving Theorems 2 (iii) and 3 (iii) and Theorem 2 (iv).
Proof of Theorem 2 (iii) and Theorem 3 (iii)
By the conditions in part (iii) of these two theorems, we see that
if
if
and the assertion follows. ∎
5 Case of fourth secant variety of Veronese embedding
In this section, we aim to prove Theorem 5 as an investigation of the singular loci of the fourth secant variety (i.e.,
5.1 Equations by Young flattening
In [24], another source of equations for secant varieties of Veronese varieties was introduced via the so-called Young flattening. Here we briefly review the construction of a certain type of Young flattening and use it to compute the conormal space of a given form.
Let
which is obtained by first embedding
For any
Let
and if we take
which shows
Thus, from
We can also use this Young flattening to compute conormal space of secant varieties of Veronese.
Let
in
Then we have
where the right-hand side is to be understood as the image of the multiplication
Proof
This proposition follows directly from the same idea as Proposition 23 by applying it to a linear embedding
Since
and not in the previous secants of the same Segre variety, this is straightforward from the proof of Proposition 23 (i.e., the case
5.2 Singularity and non-singularity
Using Proposition 28, we have the non-singularity of
Theorem 29 (From full-secant locus)
Let
Proof
First, note that, for every 𝑓 in the statement, there exists a unique 4-dimensional subspace 𝑈 such that
whose fibers
In case of
due to Landsberg–Teitler (see [22, Theorem 10.9.3.1] or [25, Theorem 10.4]) such as
Case (i)
Case (ii)
defined in (5.1).
For simplicity, we will denote this type of Young flattening by 𝜙 throughout the proof.
Then
for some nonzero
For
and the corresponding four points can be chosen as
in
For any
where
is an ideal in
for
for any
which implies that, by Proposition 28,
Hence
Case (iii)
for some nonzero
in
produce a desired 4-dimensional subspace
Similarly, for the case of
which is generated by
using a subspace
which means that
Case (iv)
and
in
For
and
Then one can check that
(here, the underline means the leading term with respect to the lexicographic order).
Say
which shows that the Hilbert function of 𝐼 can be computed as
This implies that
which means that
Case (v) The final form
in
For each
and
Then one can check that
Note that
As a direct consequence of the main results in the paper, we also obtain the following corollary on the (non-)singularity of subsecant loci in the fourth secant variety.
Corollary 30 (From subsecant loci)
Let
A general point in
Proof
As
(i) For
(ii) This is given by Theorem 1 for the case
We add some remarks on Corollary 30.
(a) For
(b) As pointed out in Example 27, a singularity can occur at a special point in
even for
Finally, we end this section by listing cases in which the same nice description for the singular locus of
Let 𝑉 be an
which is an irreducible locus of dimension
and is equal to the maximum subsecant locus
Proof
For case (i), the assertion is immediate since it corresponds to symmetric matrices.
In case (ii), we draw the conclusion from the fact that
For the remaining cases, we first claim that, for any
We note that the right-hand side of (5.5) is an irreducible and closed subvariety of
sending each subspace 𝐿 of dimension
For case (iii), by [16, Theorem 2.1, Remark 2.4 (a), and Corollary 2.11] and Theorem 1, and for case (iv), by Theorem 5, we know that
which can also be written as
In both cases (iii) and (iv), we have
which is irreducible and can be described as written in the statement. The formula for the dimension is immediate from dimension counting. ∎
6 Concluding remark
So far, we have reported results on singular loci of
As we mentioned in the introduction, each point
Issue (i) is expected to be very complicated because, at some special point, a singularity can also occur even for a low 𝑘 as shown in Example 27 (in fact, we can generate more examples using a similar idea).
For the points in the subsecant loci, in general, one could not hope to find some nice “normal forms” and the situation is expected to be wild (in other words, the subsecant loci may not be covered with finitely many nice families of
Suppose that
where
Proof
Suppose that inclusion (6.1) does not hold.
Then, taking
and by the assumption on 𝑘, we know that
On the other hand, since
and
Moreover, we may assume
and changing coordinates
which corresponds to the tailing “∗” part in (2.13) (recall that
which is contrary to
Remark 34 (Partial subsecant locus)
This new singular locus
in (6.1) can be seen as a “partial version” of subsecant locus in this paper.
In particular, it contains the 𝑚-subsecant variety
Therefore, one proper question on the singular locus of
Let
Note that the answer to Question 35 is affirmative in cases of
Let us consider
Finally, we would like to remark that the approach based on the same spirit of trichotomy pattern of (non-)singularity on subsecant loci still can be applied to the study of singular loci of higher secant varieties of other classical varieties such as Segre embeddings, Segre–Veronese varieties and Grassmannians. For instance, we can have a conjectural result like the following.
For
and denote
This can recover the result on the singular locus of the secant varieties of Segre embeddings [28, Corollary 7.17] for
We plan to deal with these cases in a forthcoming paper.
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 22K03236
Funding source: National Research Foundation of Korea
Award Identifier / Grant number: 2017R1E1A1A03070765
Award Identifier / Grant number: 2021R1F1A104818611
Funding statement: The first named author was supported by JSPS KAKENHI Grant Number 22K03236. The second named author was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT, No. 2017R1E1A1A03070765 and No. 2021R1F1A104818611).
Acknowledgements
The authors would like to give thank to the Korea Institute for Advanced Study (KIAS) for inviting them and giving a chance to start the project. We are grateful to Giorgio Ottaviani for a helpful comment on Example 6 and to Luca Chiantini and Jarek Buczyński for related discussions as well. The first author also wishes to express his gratitude to Hajime Kaji and Hiromichi Takagi for useful discussions. The computer algebra package Macaulay2 [15] was quite useful for many computations. Finally, we would like to thank the anonymous referees very much for their careful reading, pointing out some error in the first version, and many suggestions which helped us to improve the final exposition of the paper.
References
[1] H. Abo, G. Ottaviani and C. Peterson, Non-defectivity of Grassmannians of planes, J. Algebraic Geom. 21 (2012), no. 1, 1–20. Search in Google Scholar
[2] J. Alexander and A. Hirschowitz, Polynomial interpolation in several variables, J. Algebraic Geom. 4 (1995), no. 2, 201–222. Search in Google Scholar
[3] E. Angelini and L. Chiantini, On the identifiability of ternary forms, Linear Algebra Appl. 599 (2020), 36–65. Search in Google Scholar
[4] M. Brion and S. Kumar, Frobenius splitting methods in geometry and representation theory, Progr. Math. 231, Birkhäuser, Boston 2005. Search in Google Scholar
[5] W. Buczyńska and J. Buczyński, Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes, J. Algebraic Geom. 23 (2014), no. 1, 63–90. Search in Google Scholar
[6] J. Buczyński, A. Ginensky and J. M. Landsberg, Determinantal equations for secant varieties and the Eisenbud–Koh–Stillman conjecture, J. Lond. Math. Soc. (2) 88 (2013), no. 1, 1–24. Search in Google Scholar
[7] L. Chiantini, G. Ottaviani and N. Vannieuwenhoven, An algorithm for generic and low-rank specific identifiability of complex tensors, SIAM J. Matrix Anal. Appl. 35 (2014), no. 4, 1265–1287. Search in Google Scholar
[8] L. Chiantini, G. Ottaviani and N. Vannieuwenhoven, On generic identifiability of symmetric tensors of subgeneric rank, Trans. Amer. Math. Soc. 369 (2017), no. 6, 4021–4042. Search in Google Scholar
[9] J. Choe and S. Kwak, A matryoshka structure of higher secant varieties and the generalized Bronowski’s conjecture, Adv. Math. 406 (2022), Article ID 108526. Search in Google Scholar
[10] C. Ciliberto and F. Russo, Varieties with minimal secant degree and linear systems of maximaldimension on surfaces, Adv. Math. 200 (2006), no. 1, 1–50. Search in Google Scholar
[11] K. Furukawa and K. Han, On the prime ideals of higher secant varieties of Veronese embeddings of small degrees, preprint (2024), https://arxiv.org/abs/2410.00652. Search in Google Scholar
[12] K. Furukawa and A. Ito, Gauss maps of toric varieties, Tohoku Math. J. (2) 69 (2017), no. 3, 431–454. Search in Google Scholar
[13] V. Galgano and R. Staffolani, Identifiability and singular locus of secant varieties to Grassmannians, Collect. Math. (2024), 10.1007/s13348-023-00429-1. 10.1007/s13348-023-00429-1Search in Google Scholar
[14] F. Galuppi and M. Mella, Identifiability of homogeneous polynomials and Cremona transformations, J. reine angew. Math. 757 (2019), 279–308. Search in Google Scholar
[15] D. R. Grayson and M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/. Search in Google Scholar
[16] K. Han, On singularities of third secant varieties of Veronese embeddings, Linear Algebra Appl. 544 (2018), 391–406. Search in Google Scholar
[17] J. Harris, Algebraic geometry, Grad. Texts in Math. 133, Springer, New York 1992. Search in Google Scholar
[18] R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, New York 1977. Search in Google Scholar
[19] A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Math. 1721, Springer, Berlin 1999. Search in Google Scholar
[20] V. Kanev, Chordal varieties of Veronese varieties and catalecticant matrices, Algebr. Geom. 9 (1999), 1114–1125. Search in Google Scholar
[21] M. A. Khadam, M. Michałek and P. Zwiernik, Secant varieties of toric varieties arising from simplicial complexes, Linear Algebra Appl. 588 (2020), 428–457. Search in Google Scholar
[22] J. M. Landsberg, Tensors: Geometry and applications, Grad. Stud. Math. 128, American Mathematical Society, Providence 2012. Search in Google Scholar
[23] J. M. Landsberg, Geometry and complexity theory, Cambridge Stud. Adv. Math. 169, Cambridge University, Cambridge 2017. Search in Google Scholar
[24] J. M. Landsberg and G. Ottaviani, Equations for secant varieties of Veronese and other varieties, Ann. Mat. Pura Appl. (4) 192 (2013), no. 4, 569–606. Search in Google Scholar
[25] J. M. Landsberg and Z. Teitler, On the ranks and border ranks of symmetric tensors, Found. Comput. Math. 10 (2010), no. 3, 339–366. Search in Google Scholar
[26] H. Matsumura, Commutative ring theory, 2nd ed., Cambridge Stud. Adv. Math. 8, Cambridge University, Cambridge 1989. Search in Google Scholar
[27] M. Mella, Singularities of linear systems and the Waring problem, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5523–5538. Search in Google Scholar
[28] M. Michałek, L. Oeding and P. Zwiernik, Secant cumulants and toric geometry, Int. Math. Res. Not. IMRN 2015 (2015), no. 12, 4019–4063. Search in Google Scholar
[29] F. Russo, On the geometry of some special projective varieties, Lect. Notes Unione Mat. Ital. 18, Springer, Cham 2016. Search in Google Scholar
[30] F. L. Zak, Tangents and secants of algebraic varieties, Transl. Math. Monogr. 127, American Mathematical Society, Providence 1993. Search in Google Scholar
© 2025 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Frontmatter
- K-moduli of Fano threefolds and genus four curves
- Conformally covariant boundary operators and sharp higher order CR Sobolev trace inequalities on the Siegel domain and complex ball
- Dynamique analytique sur 𝐙. II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel
- Continuous sparse domination and dimensionless weighted estimates for the Bakry–Riesz vector
- Entire hypersurfaces of constant scalar curvature in Minkowski space
- On the singular loci of higher secant varieties of Veronese embeddings
- A proof of Vishik’s nonuniqueness theorem for the forced 2D Euler equation
- The width of embedded circles
Articles in the same Issue
- Frontmatter
- K-moduli of Fano threefolds and genus four curves
- Conformally covariant boundary operators and sharp higher order CR Sobolev trace inequalities on the Siegel domain and complex ball
- Dynamique analytique sur 𝐙. II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel
- Continuous sparse domination and dimensionless weighted estimates for the Bakry–Riesz vector
- Entire hypersurfaces of constant scalar curvature in Minkowski space
- On the singular loci of higher secant varieties of Veronese embeddings
- A proof of Vishik’s nonuniqueness theorem for the forced 2D Euler equation
- The width of embedded circles
