Scheme-theoretic Whitney conditions and applications to tangency of projective varieties

TL;DR

This paper introduces a scheme-theoretic version of Whitney condition a, linking it to the smoothness of the dual variety, and applies it to problems in tangency, osculating planes, and algebraic boundaries in projective and real algebraic geometry.

Contribution

It develops a scheme-theoretic framework for Whitney conditions and demonstrates its applications to tangency and boundary problems in algebraic geometry.

Findings

Proves a Bertini-type theorem for osculating planes of space curves.

Establishes a connection between Whitney conditions and dual variety smoothness.

Generalizes a theorem on algebraic boundaries of real varieties.

Abstract

We investigate a scheme-theoretic variant of Whitney condition a. If X is a projec-tive variety over the field of complex numbers and Y $\subset$ X a subvariety, then X satisfies generically the scheme-theoretic Whitney condition a along Y provided that the pro-jective dual of X is smooth. We give applications to tangency of projective varieties over C and to convex real algebraic geometry. In particular, we prove a Bertini-type theorem for osculating plane of smooth complex space curves and a generalization of a Theorem of Ranestad and Sturmfels describing the algebraic boundary of an affine compact real variety.

Equations

$$X_r^{\ast }:=\{H^{\perp }\in X^{\ast },\text{such that}\dim ⟨(H\cap X{)}_{tan}⟩\ge r\}.$$

$$\left\{\frac{\partial F_H}{\partial x_1},\cdots ,\frac{\partial F_H}{\partial x_n}\right\}.$$

$$x_j\frac{\partial F_H}{\partial x_i}=0,$$

$$\dim ⟨(H\cap X{)}_{tan}⟩\ge N-n(n+1).$$

$${\tilde{X}}_r^{\ast }:=\{H^{\perp }\in X^{\ast },\text{such that}\dim ⟨(H\cap X{)}_{tan}^{red}⟩\ge r\}.$$

$${\overline{X}}_r^{\ast }:=\{H^{\perp }\in X^{\ast },\text{such that}\dim ⟨T_{(H\cap X{)}_{tan}}⟩\ge r\}.$$

$$\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}{I}_X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\tilde{X}}_r^{\ast }:=\{H^{\perp }\in X^{\ast },\text{such that}\dim ⟨(H\cap X{)}_{tan}^{red}⟩\ge r\}.$$

$$I_{\hat{X}}=I_{\widehat{X^{\ast }}}.$$

$$\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}{I}_X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$Z_r=\{H^{\perp }\in \widehat{X^{\ast }},\text{such that}\dim ⟨\{q(p^{-1}(H)){\}}_{red}⟩\ge r+1\}.$$

$$\displaystyle\langle Z_{red}\rangle\subset\Big{\langle}\overline{T_{Z^{\bullet}}}\Big{\rangle}\subset\langle Z\rangle$$

$$\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}{I}_X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\omega :{\Omega }_{I_X,q^{-1}(y{)}^{\bullet }}\times {\Omega }_{I_X,q^{-1}(y{)}^{\bullet }}⟶{\mathcal{O}}_{q^{-1}(y{)}^{\bullet }}$$

$$0⟶q^{\ast }{\Omega }_{Y,y}⟶{\Omega }_{q^{-1}(U),q^{-1}(y)}.$$

$${\Omega }_{I_X,q^{-1}(y)}⟶{\Omega }_{q^{-1}(U),q^{-1}(y)}⟶0.$$

$$0⟶p^{\ast }{\Omega }_{X^{\ast },p(q^{-1}(y){)}^{\bullet }}⟶{\Omega }_{I_X,q^{-1}(y{)}^{\bullet }}.$$

$${\Omega }_{X^{\ast },p(q^{-1}(y){)}^{\bullet }}⟶{\Omega }_{p(q^{-1}(y){)}^{\bullet }}⟶0.$$

$$\omega :q^{\ast }{\Omega }_{Y,y}\times p^{\ast }{\Omega }_{p(q^{-1}(y){)}^{\bullet }}⟶{\mathcal{O}}_{p(q^{-1}(y){)}^{\bullet }}$$

$$\omega :q^{\ast }{T}_{Y,y}\times p^{\ast }{T}_{p(q^{-1}(y){)}^{\bullet },z}⟶\mathbb{C}$$

$$\Big{\langle}\overline{T_{p(q^{-1}(y))^{\bullet}}}\Big{\rangle}\subset T_{Y,y}^{\perp}.$$

$$\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}{I}_X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Big{\langle}\overline{T_{p(q^{-1}(y)),X^{*}_{smooth}}}\Big{\rangle}\subset T_{Y,y}^{\perp}.$$

$${\overline{X}}_r^{\ast }:=\{H^{\perp }\in X^{\ast },\text{such that}\dim ⟨T_{(H\cap X{)}_{tan}}⟩\ge r\}.$$

$$\dim X^{\ast }(1)\le N-3.$$

$$\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}{I}_X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$p(q^{-1}(X^{\ast }(1)))=X,$$

$$⟨C_H^{extra}⟩\subset T_{(H\cap C{)}_{tan},c_H},$$

Full text

1. 1 Introduction

2. 2 Variations on the Whitney condition $\displaystyle(a)$

3. 2.1 Whitney condition $\displaystyle(a)$

4. 2.2 Scheme-theoretic Whitney condition $\displaystyle(a)$

5. 3 Applications of the scheme-theoretic Whitney conditions

6. 3.1 Classification of projective surfaces with a maximal family of maximally tangent hyperplanes

7. 3.2 A Bertini-type Theorem for osculating planes of space curves

8. 3.3 The algebraic boundary of an affine compact real variety

1 Introduction

Let $\displaystyle X\subset\mathbb{R}^{n}$ be a non-degenerate compact convex body, whose interior contains $\displaystyle 0$. Denote by $\displaystyle X^{*}\subset(\mathbb{R}^{n})^{*}$ the dual body. Let $\displaystyle x\in\partial X$, the set $\displaystyle x^{\perp}\cap X^{*}$ is called the exposed face of $\displaystyle X^{*}$ with respect to $\displaystyle x^{\perp}$. A point $\displaystyle x\in\partial X$ is said to be an $\displaystyle r$-singular point if the exposed face of $\displaystyle X^{*}$ with respect to $\displaystyle x^{\perp}$ is of dimension at least $\displaystyle r$. In[AK52], Anderson and Klee give a sharp bound on the Hausdorff dimension of the set of $\displaystyle r$-singular points of a convex body. Namely, we have the:

Theorem 1.0.1** ([AK52])**

Let $\displaystyle X\subset\mathbb{R}^{n}$ be a non-degenerate compact convex body, whose interior contains $\displaystyle 0$. The set of $\displaystyle r$-singular points of $\displaystyle X$ has Hausdorff dimension at most $\displaystyle n-r-1$.

Following work of Ranestad and Sturmfels [RS11], there have been some interest to know whether a version of Theorem 1.0.1 could possibly hold in complex algebraic geometry. Namely, I conjectured in [Abu] the following:

Conjecture 1.0.2** ([Abu])**

Let $\displaystyle X\subset\mathbb{P}^{N}$ be a smooth complex projective variety. Denote by $\displaystyle X^{*}_{r}$ the variety:

[TABLE]

Then, we have $\displaystyle\dim X^{*}_{r}\leq N-r-1$.

In the above conjecture, $\displaystyle(H\cap X)_{tan}$ is the tangency scheme of $\displaystyle H$ along $\displaystyle X$ and $\displaystyle\langle(H\cap X)_{tan}\rangle$ is the linear span of this tangency scheme. This conjecture is wrong as shows the following example which was communicated to me by Voisin:

Example 1.0.3

Let $\displaystyle X=v_{d}(\mathbb{P}^{n})\subset\mathbb{P}(S^{d}\mathbb{C}^{n+1})=\mathbb{P}^{N}$, with $\displaystyle N={n+d\choose d}-1$. Hyperplane sections of $\displaystyle X$ are isomorphic to hypersurfaces of degree $\displaystyle d$ in $\displaystyle\mathbb{P}^{n}$. Hence, the variety $\displaystyle X^{*}\subset(\mathbb{P}^{N})^{*}$ parametrizes singular hypersurfaces of degree $\displaystyle d$ in $\displaystyle\mathbb{P}^{n}$ (the variety $\displaystyle X^{*}$ is known as the discriminant). We focus on singular hypersurfaces which are cones over hypersurfaces of degree $\displaystyle d$ in $\displaystyle\mathbb{P}^{n-1}$. A trivial count of dimension shows that there is a $\displaystyle{d+n-1\choose d}+n-1$-dimensional family of such cones (choose a hypersurface of degree $\displaystyle d$ in $\displaystyle\mathbb{P}^{n-1}$ and then choose the vertex of the cone in $\displaystyle\mathbb{P}^{n}$.)

We denote by $\displaystyle X^{*}_{cone}$ the closure of the subset of $\displaystyle X^{*}$ which consists of points $\displaystyle H^{\perp}\in X^{*}$ such that $\displaystyle H\cap X$ is a cone over a hypersurface of degree $\displaystyle d$ in $\displaystyle\mathbb{P}^{n-1}$. Let $\displaystyle H^{\perp}$ be generic in $\displaystyle X^{*}_{cone}$. We will give a lower bound on the linear span of $\displaystyle(H\cap X)_{tan}$. We fix coordinates (say $\displaystyle x_{0},\cdots,x_{n}$) on $\displaystyle\mathbb{P}^{n}$, such that the vertex of $\displaystyle H\cap X$ is the point $\displaystyle[1:0:\cdots:0]$. We denote by $\displaystyle F_{H}$ the degree $\displaystyle d$ homogeneous polynomial in the $\displaystyle x_{i}$ corresponding to $\displaystyle H\cap X$. Since $\displaystyle H\cap X$ is a cone with vertex $\displaystyle[1:0:\cdots:0]$, $\displaystyle F_{H}$ only depends from $\displaystyle x_{1},\cdots,x_{n}$. The tangency locus $\displaystyle(H\cap X)_{tan}$ is defined in $\displaystyle X$ by the vanishing of the partial derivatives:

[TABLE]

(we do a slight abuse of notations here as we consider the scheme defined by the ideal $\displaystyle\left(\dfrac{\partial F_{H}}{\partial x_{1}},\cdots,\dfrac{\partial F_{H}}{\partial x_{n}}\right)$ as a subscheme of $\displaystyle X$ and not a subscheme of $\displaystyle\mathbb{P}^{n}$).

Furthermore, as $\displaystyle x_{0},\cdots,x_{n}$ is a system of coordinates for $\displaystyle\mathbb{P}^{n}$, we know that the ideals $\displaystyle\left(\dfrac{\partial F_{H}}{\partial x_{i}}\right)$ and $\displaystyle\left(x_{0}\dfrac{\partial F_{H}}{\partial x_{i}},\cdots,x_{n}\dfrac{\partial F_{H}}{\partial x_{i}}\right)$ are equal for all $\displaystyle i$. Hence, the subscheme of $\displaystyle X$ defined by the ideal $\displaystyle\left(\dfrac{\partial F_{H}}{\partial x_{1}},\cdots,\dfrac{\partial F_{H}}{x_{n}}\right)$ is equal to the subscheme of $\displaystyle X$ defined by the ideal $\displaystyle\left(x_{i}\dfrac{\partial F_{H}}{\partial x_{j}}\right)_{i\in\{0\cdots n\},j\in\{1,\cdots n\}}$. We deduce that the linear span of $\displaystyle(H\cap X)_{tan}$ in $\displaystyle\mathbb{P}^{N}$ is defined by the equations:

[TABLE]

for all $\displaystyle i\in\{1,\cdots,n\}$ and all $\displaystyle j\in\{0,\cdots n\}$. As a consequence, we find:

[TABLE]

We have seen that $\displaystyle\dim X^{*}_{cone}={d+n-1\choose d}+n-1$. But $\displaystyle{d+n-1\choose d}+n-1>N-(N-n(n+1))-1=n(n+1)-1$ as soon as $\displaystyle d\geq 4$ if $\displaystyle n\geq 3$. So that $\displaystyle X^{*}_{cone}$ is a counter-example to Conjecture 1.0.2 as soon as $\displaystyle d\geq 4$ and $\displaystyle n\geq 3$.

For the applications found in [RS11], only the set-theoretic version Conjecture 1.0.2 is needed. It is stated in [Abu] that this set-theoretic version is true and we have:

Theorem 1.0.4

Let $\displaystyle X\subset\mathbb{P}^{N}$ be a projective variety. Denote by $\displaystyle\tilde{X}^{*}_{r}$ the variety:

[TABLE]

Then, we have $\displaystyle\dim\tilde{X}^{*}_{r}\leq N-r-1$.

Here $\displaystyle\langle(H\cap X)^{red}_{tan}\rangle$ is the linear span of the reduced tangency locus of $\displaystyle H$ with $\displaystyle X$. A complicated proof of this Theorem was announced in [Abu]. In section 2 of this paper, we will give a slick proof of Theorem 1.0.4 based on the fact that Whitney condition $\displaystyle(a)$ is stratifying.

It was however noticed by Peskine and Zak that the Conjecture 1.0.2 is intriguing precisely because it would imply the existence of unexpected bounds on the dimensions of families of osculating hyperplanes. With this perspective in mind, Theorem 1.0.4 seems a bit less appealing as it says nothing about osculating hyperplanes. In fact, in order to prove a result which would have a similar flavor as Conjecture 1.0.2, one has to replace the linear span of the tangency locus by the linear span of the union of the tangent spaces to the tangency locus. The difference between these two objects might look quite thin at first sight. Indeed, both objects coincide if the tangency locus is reduced : this is one of the reasons which explain why the erroneous conjecture 1.0.2 was made in the first place. The example of the $\displaystyle d$-th Veronese embedding of $\displaystyle\mathbb{P}^{n}$ in $\displaystyle\mathbb{P}(S^{d}\mathbb{C}^{n+1})$ shows that these two objects may differ in general.

Example 1.0.5

Let $\displaystyle X=v_{d}(\mathbb{P}^{n})\subset\mathbb{P}(S^{d}\mathbb{C}^{n+1})=\mathbb{P}^{N}$. Let $\displaystyle H^{\perp}\in X^{*}$ such that $\displaystyle H\cap X$ is a cone over a smooth hypersurface in $\displaystyle\mathbb{P}^{n-1}$, with vertex denoted by $\displaystyle x$. The hyperplane $\displaystyle H$ is tangent to $\displaystyle X$ only at $\displaystyle x$. Furthermore, as explained in example 1.0.3, we have $\displaystyle\dim\langle(H\cap X)_{tan}\rangle\geq\binom{n+d}{d}-1-n(n+1)$. On the other hand, we have $\displaystyle T_{(H\cap X)_{tan},x}\subset T_{X,x}$, so that $\displaystyle\dim\langle T_{(H\cap X)_{tan}}\rangle\leq n$. The quantities $\displaystyle\binom{n+d}{d}-1-n(n+1)$ and $\displaystyle n$ differ as soon as $\displaystyle d\geq 4$ and $\displaystyle n\geq 3$.

The linear span of the union of the tangent spaces to the tangency locus seems to be the correct linear span to consider as soon as stratification of the projective dual is concerned. Indeed, the main result of this paper is the:

Theorem 1.0.6

Let $\displaystyle X\subset\mathbb{P}^{N}$ be a smooth projective variety. Denote by $\displaystyle\overline{X}^{*}_{r}$ the variety:

[TABLE]

Then, we have $\displaystyle\dim\overline{X}^{*}_{r}\leq N-r-1$.

Here $\displaystyle\langle T_{(H\cap X)_{tan}}\rangle$ is the linear span of the union of the tangent spaces to the tangency scheme of $\displaystyle H$ with $\displaystyle X$. The proof of this result is based on the stratifying property of a certain scheme-theoretic version of Whitney condition $\displaystyle(a)$ for the pair $\displaystyle(X^{*},Y)$ ($\displaystyle Y$ a subvariety of $\displaystyle X^{*}$), provided that $\displaystyle X$ is smooth. The smoothness assumption can be dropped if one defines $\displaystyle\langle T_{(H\cap X)_{tan}}\rangle$ as the linear span of the Zariski closure of the union of the tangent spaces to the tangency scheme of $\displaystyle H$ with $\displaystyle X$ at smooth points of $\displaystyle X$. We will come back to this more technical statement in section $\displaystyle 2$ of the paper.

In section $\displaystyle 3$, we will give applications of this result to both the study of tangency of projective varieties defined over the field of complex numbers and to convex real algebraic geometry. In particular, we will prove a Bertini-type result for osculating planes of smooth complex space curves (see Theorem 3.2.1) and a generalization of a result due to Ranestad and Sturmfels describing the algebraic boundary of an affine compact real variety (see Theorem 3.3.1).

Acknowledgments     I am very grateful to Christian Peskine and Fyodor Zak for the numerous and fruitful discussions we have had on Conjecture 1.0.2. It was their insight that the the linear span of the union of the tangent spaces to the tangency scheme should replace the linear span of the tangency scheme in the statement of Conjecture 1.0.2. I am also indebted to Claire Voisin for sharing with me her counter-example to Conjecture 1.0.2.

2 Variations on the Whitney condition $\displaystyle(a)$

2.1 Whitney condition $\displaystyle(a)$

In this section we gather some elementary facts about Whitney condition $\displaystyle(a)$ and we use them to prove the set-theoretic version of Conjecture 1.0.2. Recall the definition of Whitney condition $\displaystyle(a)$ (see [Whi65] or [Tei82] for instance)

Definition 2.1.1

Let $\displaystyle X\subset\mathbb{C}^{n}$ be an (algebraic) variety, $\displaystyle Y\subset X$ a subvariety of $\displaystyle X$ and $\displaystyle y\in Y$. We say that the pair $\displaystyle(X,Y)$ satisfies the Whitney condition $\displaystyle(a)$ at $\displaystyle y$ if for any sequence $\displaystyle\{x_{n}\}$ of points of $\displaystyle X_{smooth}$ converging to $\displaystyle y$ such that the sequence of tangent spaces $\displaystyle T_{X,x_{n}}$ converges (in the appropriate Grassmannian), the limit tangent space $\displaystyle\lim T_{X,x_{n}}$ contains $\displaystyle T_{Y,y}$.

With notations as in definition 2.1.1, the Whitney condition $\displaystyle(a)$ is easily seen to be equivalent to the following : for any sequence $\displaystyle\{x_{n}\}$ of points of $\displaystyle X_{smooth}$ converging to $\displaystyle y$ and any converging sequence of hyperplanes $\displaystyle\{H_{n}\}$ which are tangent to $\displaystyle X$ at $\displaystyle x_{n}$, the limiting hyperplane $\displaystyle\lim H_{n}$ contains $\displaystyle T_{Y,y}$. This definition may then be further reformulated as follows:

Proposition 2.1.2

Let $\displaystyle X\subset\mathbb{C}^{N+1}$ be an algebraic variety and $\displaystyle Y\subset X$ a subvariety with a marked point $\displaystyle y\in Y$. Denote by $\displaystyle I_{X}\subset\mathbb{C}^{N+1}\times(\mathbb{C}^{N+1})^{*}$ the Zariski closure of the total space of $\displaystyle N^{*}_{X_{smooth}/\mathbb{C}^{N+1}}$ over $\displaystyle X_{smooth}$. We denote by

[TABLE]

the canonical projections. The pair $\displaystyle(X,Y)$ satisfies the Whitney condition $\displaystyle(a)$ at $\displaystyle y$ if and only if $\displaystyle\{q(p^{-1}(y))\}_{red}\subset T_{Y,y}^{\perp}$.

Here $\displaystyle\{q(p^{-1}(y))\}_{red}$ is the reduced scheme underlying $\displaystyle q(p^{-1}(y))$.

Proof :

$\displaystyle\blacktriangleright\;$ By definition of $\displaystyle I_{X}$, the reduced scheme $\displaystyle\{q(p^{-1}(y))\}_{red}$ is the set of limits of converging sequences of hyperplanes tangent to $\displaystyle X_{smooth}$ at sequences of points which converge to $\displaystyle y$. Hence, the pair $\displaystyle(X,Y)$ satisfies the Whitney condition $\displaystyle(a)$ at $\displaystyle y$ if and only if $\displaystyle\{q(p^{-1}(y))\}_{red}\subset T_{Y,y}^{\perp}$. $\displaystyle\blacktriangleleft$

The Whitney condition $\displaystyle(a)$ is an interesting condition because it is stratifying. More precisely, we have:

Theorem 2.1.3** ([Whi65])**

Let $\displaystyle X\subset\mathbb{C}^{N+1}$ be an algebraic variety and let $\displaystyle Y\subset X$ be a subvariety. There exists a non-empty Zariski open subset $\displaystyle U\subset Y$ such that the pair $\displaystyle(X,Y)$ satisfies the Whitney condition $\displaystyle(a)$ for all $\displaystyle y\in U$.

This result was originally proved by Whitney [Whi65]. We refer to [Tei82] for a quick proof in the algebraic setting. As a consequence of this stratifying property, one can immediately deduce the following:

Theorem 2.1.4

Let $\displaystyle X\subset\mathbb{P}^{N}$ be a projective variety. Denote by $\displaystyle\tilde{X}^{*}_{r}$ the variety:

[TABLE]

Then, we have $\displaystyle\dim\tilde{X}^{*}_{r}\leq N-r-1$.

Proof :

$\displaystyle\blacktriangleright\;$ Let $\displaystyle\widehat{X}\subset\mathbb{C}^{N+1}$ be the affine cone over $\displaystyle X$. Let $\displaystyle I_{\widehat{X}}\subset\mathbb{C}^{N+1}\times(\mathbb{C}^{N+1})^{*}$ be the Zariski closure of the total space of $\displaystyle N^{*}_{\widehat{X}_{smooth}/\mathbb{C}^{N+1}}$ over $\displaystyle\widehat{X}_{smooth}$. We denote by $\displaystyle\widehat{X^{*}}$ the image of the projection of $\displaystyle I_{X}$ on $\displaystyle(\mathbb{C}^{N+1})^{*}$ (note that $\displaystyle\widehat{X^{*}}$ is the affine cone over the projective dual of $\displaystyle X$). If $\displaystyle I_{\widehat{X^{*}}}\subset\mathbb{C}^{N+1}\times(\mathbb{C}^{N+1})^{*}$ is the Zariski closure of the total space of $\displaystyle N^{*}_{\widehat{X^{*}}_{smooth}/(\mathbb{C}^{N+1})^{*}}$ over $\displaystyle\widehat{X^{*}}_{smooth}$, the Lagrangian duality (also called Reflexivity) asserts that:

[TABLE]

Denote by :

[TABLE]

the canonical projections. Let $\displaystyle Z_{r}$ be the affine cone over $\displaystyle\tilde{X}^{*}_{r}$. By definition,

[TABLE]

Let $\displaystyle H^{\perp}$ be a generic point in $\displaystyle Z_{r}$. By Theorem 2.1.3, we know that $\displaystyle\langle\{q(p^{-1}(H^{\perp}))\}_{red}\subset T_{Z_{r},H^{\perp}}^{\perp}$. This implies that $\displaystyle\dim Z_{r}\leq N+1-(r+1)$. As a consequence, we conclude that $\displaystyle\dim\tilde{X}^{*}_{r}\leq N-1-r$. $\displaystyle\blacktriangleleft$

2.2 Scheme-theoretic Whitney condition $\displaystyle(a)$

In this section we will explore a scheme-theoretic variant of Whitney condition $\displaystyle(a)$ for varieties which are afinne cones over projective varieties. We prove that this condition is stratifying if the projective dual of the ambient variety is smooth. In the following, if $\displaystyle Z\subset\mathbb{C}^{N+1}$ is the affine cone over a closed projective scheme say $\displaystyle\mathbb{P}(Z)\subset\mathbb{P}^{N}$, we denote by $\displaystyle Z^{\bullet}$ the scheme $\displaystyle Z\backslash\{0\}$ and we let \displaystyle\Big{\langle}\overline{T_{Z^{\bullet}}}\Big{\rangle} be the linear span of the Zariski closure of the the union of the tangent spaces to $\displaystyle Z^{\bullet}$. Notice that we have the chain of inclusions:

[TABLE]

and these inclusions are all equalities if $\displaystyle Z$ is reduced. Example 1.0.5 shows that they can be both strict if $\displaystyle Z$ is non reduced. More genereally, if $\displaystyle Z\subset X\subset\mathbb{C}^{N+1}$ is a closed subscheme of $\displaystyle X$ in $\displaystyle\mathbb{C}^{N+1}$, we denote by \displaystyle\Big{\langle}\overline{T_{Z,X_{smooth}}}\Big{\rangle}, the linear span in $\displaystyle\mathbb{C}^{N+1}$ of the Zariski closure of the union of the tangent spaces to $\displaystyle Z$ at smooth points of $\displaystyle X$.

Definition 2.2.1

Let $\displaystyle X\subset\mathbb{C}^{N+1}$ be the affine cone over a projective variety and let $\displaystyle Y\subset X$ be the affine cone over a projective subvariety of $\displaystyle\mathbb{P}(X)$. Let $\displaystyle y\in Y$ and let $\displaystyle I_{X}\subset\mathbb{C}^{N+1}\times(\mathbb{C}^{N+1})^{*}$ be the closure of $\displaystyle N^{*}_{X_{smooth}/\mathbb{C}^{N+1}}$ in $\displaystyle\mathbb{C}^{N+1}\times(\mathbb{C}^{N+1})^{*}$. We denote by

[TABLE]

the canonical projections. We say that the pair $\displaystyle(X,Y)$ satisfies the scheme-theoretic Whitney condition $\displaystyle(a)$ at $\displaystyle y$ if \displaystyle\Big{\langle}\overline{T_{p(q^{-1}(y))^{\bullet}}}\Big{\rangle}\subset T_{Y,y}^{\perp}.

Remark 2.2.2

It is obvious that the pair $\displaystyle(X,Y)$ satifies the Whitney condition (a) at $\displaystyle y$ if it satisfies the scheme-theoretic Whitney conditions (a) at $\displaystyle y$. The converse is false. 2. 2.

Since $\displaystyle X$ is the affine cone over a projective variety (denoted by $\displaystyle\mathbb{P}(X)$), the variety $\displaystyle X^{*}=p(I_{X})$ is the affine cone over the projective dual of $\displaystyle\mathbb{P}(X)$, which we denote by $\displaystyle\mathbb{P}(X^{*})$. For all $\displaystyle y\in Y$, with $\displaystyle y\neq 0$, the scheme $\displaystyle p(q^{-1}(y))$ is the affine cone over a projective scheme (which we denote by $\displaystyle\mathbb{P}\{p(q^{-1}(y))\}$). One readily checks that \displaystyle\Big{\langle}\overline{T_{p(q^{-1}(y)),X^{*}_{smooth}}}\Big{\rangle} is the affine cone over the linear subspace of $\displaystyle(\mathbb{P}^{N})^{*}$ spanned by the union of the (embedded in $\displaystyle(\mathbb{P}^{N})^{*}$) tangent spaces to $\displaystyle\mathbb{P}\{p(q^{-1}(y))\}$ at smooth points of $\displaystyle\mathbb{P}(X^{*})$. In particular, if $\displaystyle\mathbb{P}\{p(q^{-1}(y))\}\subset\mathbb{P}(X^{*})_{smooth}$ , then \displaystyle\Big{\langle}\overline{T_{p(q^{-1}(y)),X^{*}_{smooth}}}\Big{\rangle}=\Big{\langle}\overline{T_{p(q^{-1}(y))^{\bullet}}}\Big{\rangle} is the affine cone in $\displaystyle(\mathbb{C}^{N+1})^{*}$ over the linear subspace of $\displaystyle(\mathbb{P}^{N})^{*}$ spanned by the union of the (embedded) tangent spaces to $\displaystyle\mathbb{P}\{p(q^{-1}(y))\}$.

Now we come to the main result of this section.

Theorem 2.2.3

Let $\displaystyle X\subset\mathbb{C}^{N+1}$ be the affine cone over a projective variety and let $\displaystyle Y\subset X$ be the affine cone over a projective subvariety of $\displaystyle\mathbb{P}(X)$. We denote by :

[TABLE]

the closure of $\displaystyle N^{*}_{X_{smooth}/\mathbb{C}^{N+1}}$ in $\displaystyle\mathbb{C}^{N+1}\times(\mathbb{C}^{N+1})^{*}$ with its canonical projections. Assume that for generic $\displaystyle y\in Y$, the projectivization $\displaystyle\mathbb{P}\{p(q^{-1}(y))\}$ lies in $\displaystyle\mathbb{P}(X^{*})_{smooth}$. Then, the pair $\displaystyle(X,Y)$ satisfies generically along $\displaystyle Y$ the scheme-theoretic Whitney condition $\displaystyle(a)$.

Proof :

$\displaystyle\blacktriangleright\;$

We know that for generic $\displaystyle y\in Y$, the projectivization $\displaystyle\mathbb{P}\{p(q^{-1}(y))\}$ lies in $\displaystyle\mathbb{P}(X^{*})_{smooth}$. Hence, there exists a dense open subset $\displaystyle U\subset$Y, such that for all $\displaystyle y\in U$, the projectivized fiber $\displaystyle\mathbb{P}\{p(q^{-1}(y))\}$ lies in $\displaystyle\mathbb{P}(X^{*})_{smooth}$. Put differently, for all $\displaystyle y\in U$ and for all $\displaystyle z\in p(q^{-1}(y))$ such that $\displaystyle z\neq 0$, we have $\displaystyle z\in X^{*}_{smooth}$.

By Lagrangian duality (also called reflexivity, see [GKZ94], chapter 1), we know that $\displaystyle I_{X}=I_{X}^{*}$ is a Lagragngian subvariety of $\displaystyle\mathbb{C}^{N+1}\times\left(\mathbb{C}^{N+1}\right)^{*}$. Hence, for all $\displaystyle(x,z)\in I_{X}$ such that $\displaystyle z\in X^{*}_{smooth}$, the vector space $\displaystyle\Omega_{I_{X},(x,z)}$ is a Lagrangian subspace of $\displaystyle\mathbb{C}^{N+1}\times(\mathbb{C}^{N+1})^{*}$ with respect to the natural symplectic form (denoted by $\displaystyle\omega$) on $\displaystyle\mathbb{C}^{N+1}\times(\mathbb{C}^{N+1})^{*}$. In particular, for $\displaystyle y\in U$, the sheaf $\displaystyle\Omega_{I_{X},q^{-1}(y)^{\bullet}}$ is a Lagrangian subbundle of $\displaystyle\mathbb{C}^{N+1}\times(\mathbb{C}^{N+1})^{*}\otimes\mathscr{O}_{q^{-1}(y)^{\bullet}}$ with respect to $\displaystyle\omega$. As a consequence, the two form:

[TABLE]

is everywhere vanishing.

We consider the restriction of the projection map $\displaystyle q:q^{-1}(Y)\longrightarrow Y$. Since we are working over $\displaystyle\mathbb{C}$, up to shrinking $\displaystyle U$, one can assume that the codifferential map gives an injection $\displaystyle 0\longrightarrow q^{*}\Omega_{U}\longrightarrow\Omega_{q^{-1}(U)}$ (this is the generic smoothness Theorem). As a consequence, for all $\displaystyle y\in U$, we have an injection of sheaves:

[TABLE]

Moreover, as $\displaystyle q^{-1}(U)$ is a subscheme of $\displaystyle I_{X}$, we have a surjection:

[TABLE]

Notice that we also have an injection:

[TABLE]

Indeed, the morphism $\displaystyle p:I_{X}\longrightarrow X^{*}$ is smooth over $\displaystyle p(q^{-1}(y))^{\bullet}$ as it identifies with the fibration $\displaystyle N^{*}_{X^{*}_{smooth}/(\mathbb{C}^{N+1})^{*}}\longrightarrow X^{*}_{smooth}$. Finally, as $\displaystyle p(q^{-1}(y))^{\bullet}$ is a subscheme of $\displaystyle X^{*}$, we have a surjection:

[TABLE]

Combining equations 1, 2, 3, 4 and 5, we find that the two form:

[TABLE]

is everywhere vanishing. Taking tensor products with $\displaystyle\mathbb{C}(z)$ for any $\displaystyle z\in p(q^{-1}(y))^{\bullet}$ and then the dual, we find that the two form:

[TABLE]

is zero. This means that the vector space $\displaystyle T_{Y,y}$ is orthogonal to $\displaystyle T_{p(q^{-1}(y)),z}$ for all $\displaystyle z\in p(q^{-1}(y))^{\bullet}$. As a consequence, we get the inclusion

[TABLE]

This concludes the proof. $\displaystyle\blacktriangleleft$

As a matter of fact, it is not difficult to see that one has the more general statement:

Theorem 2.2.4** (variant of Theorem 2.2.3)**

Let $\displaystyle X\subset\mathbb{C}^{N+1}$ be the affine cone over a projective variety and let $\displaystyle Y\subset X$ be the affine cone over a projective subvariety of $\displaystyle\mathbb{P}(X)$. We denote by :

[TABLE]

the closure of $\displaystyle N^{*}_{X_{smooth}/\mathbb{C}^{N+1}}$ in $\displaystyle\mathbb{C}^{N+1}\times(\mathbb{C}^{N+1})^{*}$ with its canonical projections. Then for generic $\displaystyle y\in Y$, we have:

[TABLE]

The proof is exactly the same as for Theorem 2.2.3 and the details are left to the reader.

Corollary 2.2.5

Let $\displaystyle X\subset\mathbb{P}^{N}$ be a smooth projective variety. Denote by $\displaystyle\overline{X}^{*}_{r}$ the variety:

[TABLE]

Then, we have $\displaystyle\dim\overline{X}^{*}_{r}\leq N-r-1$.

Proof :

$\displaystyle\blacktriangleright\;$ Apply Theorem 2.2.3 to the pair $\displaystyle(X^{*},\overline{X}^{*}_{r})$, knowing that $\displaystyle(X^{*})^{*}=X$ is smooth. $\displaystyle\blacktriangleleft$

The proof of Theorem 2.2.3 shows that a variant of the scheme-theoretic Whitney condition $\displaystyle(a)$ (namely that $\displaystyle\langle\overline{T_{p(q^{-1}(y)),X^{*}_{smooth}}}\rangle\subset T_{Y,y}^{\perp}$) is always stratifying. On the other hand, the following example suggests that this variant is far from being optimal.

Example 2.2.6

Let $\displaystyle X\subset\mathbb{C}^{5}$ be the affine cone over a generic hyperplane section of $\displaystyle\mathbb{P}^{1}\times\mathbb{P}^{2}\subset\mathbb{P}^{5}$. The surface $\displaystyle\mathbb{P}(X)\subset\mathbb{P}^{4}$ is a ruled surface, for which the principal axis of the ruling is denoted by $\displaystyle\mathbb{P}(L)$. The projective dual of $\displaystyle\mathbb{P}(X)$ is a cubic hypersurface, whose singular locus is precisely $\displaystyle\mathbb{P}(L)^{\perp}$ (we refer to [Zak04] and [Abu11] for more details on this example). As $\displaystyle X$ is smooth outside $\displaystyle 0$, the pair $\displaystyle(X,L)$ obviously satisfies the scheme-theoretic Whitney condition $\displaystyle(a)$ at all $\displaystyle z\in L$ with $\displaystyle z\neq 0$. On the other hand, with notations as in Theorem 2.2.4, we have $\displaystyle p(q^{-1}(z))\subset X^{*}_{sing}$, for any $\displaystyle z\in L$. Hence $\displaystyle\langle\overline{T_{p(q^{-1}(z)),X^{*}_{smooth}}}\rangle=0$ for any $\displaystyle z\in L$ and the conclusion of Theorem 2.2.4 is void in that situation.

This suggests that there is a some room for improving Theorem 2.2.3 and Theorem 2.2.4. In particular, one could ask the following question:

Question 2.2.7

Let $\displaystyle X\subset\mathbb{C}^{N+1}$ be the affine cone over a projective variety and let $\displaystyle Y\subset X$ be the affine cone over a projective subvariety of $\displaystyle\mathbb{P}(X)$. Does the pair $\displaystyle(X,Y)$ necessarily satisfies the scheme-theoretic Whitney condition $\displaystyle(a)$ at the generic point of $\displaystyle Y$?

3 Applications of the scheme-theoretic Whitney conditions

3.1 Classification of projective surfaces with a maximal family of maximally tangent hyperplanes

The goal of this section is to give an application of Theorem 2.2.3 to the classification of smooth projective varieties with large family of maximally tangent hyperplanes.

Theorem 3.1.1

Let $\displaystyle X\subset\mathbb{P}^{N}$ be a smooth projective non-degenerate surface and $\displaystyle N\geq 5$. Denote by $\displaystyle X^{*}(1)$ the set of hyperplanes which are tangent to $\displaystyle X$ along a curve. Then

[TABLE]

If $\displaystyle\dim X^{*}(1)=N-3$, then the family of tangency curves in $\displaystyle X$ of hyperplanes parametrized by $\displaystyle X^{*}(1)$ is at least $\displaystyle 1$-dimensional. If it is at least $\displaystyle 2$-dimensional, then $\displaystyle X$ is the Veronese surface in $\displaystyle\mathbb{P}^{5}$.

Proof :

$\displaystyle\blacktriangleright\;$ Let $\displaystyle H^{\perp}\in X^{*}(1)$ be generic. Then \displaystyle\Big{\langle}T_{(H\cap X)_{tan}}\Big{\rangle}\geq 1 and we have equality if and only if $\displaystyle(H\cap X)_{tan}$ is scheme-theoretically a line. By Theorem 2.2.3, this implies that $\displaystyle\dim X^{*}(1)\leq N-2$.

Assume that $\displaystyle\dim X^{*}(1)=N-2$. For any $\displaystyle H^{\perp}\in X^{*}(1)$ we denote by $\displaystyle L_{H}$ the tangency scheme of $\displaystyle H$ along $\displaystyle X$. This is a line. Fix a general $\displaystyle H_{0}^{\perp}\in X^{*}(1)$. Assume that the dimension of the set of $\displaystyle H^{\perp}\in X^{*}(1)$ which are tangent to $\displaystyle X$ along $\displaystyle L_{H_{0}}$ is at least $\displaystyle N-3$. Since $\displaystyle N\geq 5$, we would find a $\displaystyle\mathbb{P}^{2}\subset\mathbb{P}^{N}$ which is tangent to $\displaystyle X$ along $\displaystyle L_{H_{0}}$. By Zak’s Theorem on Tangency ([Zak93]) this is impossible. Hence, the dimension of the set of $\displaystyle H^{\perp}\in X^{*}(1)$ which are tangent along $\displaystyle L_{H_{0}}$ is at most $\displaystyle N-4$. This implies that the family of lines $\displaystyle\{L_{H}\}_{H^{\perp}\in X^{*}(1)}$ is at least $\displaystyle 2$-dimensional. As a consequence, the variety $\displaystyle X$ is a $\displaystyle\mathbb{P}^{2}$ linearly embedded in $\displaystyle\mathbb{P}^{N}$. This is impossible by the non-degeneracy hypothesis. As a conclusion, we get that:

[TABLE]

Assume that $\displaystyle\dim X^{*}(1)=N-3$. Let $\displaystyle I_{X}=\mathbb{P}(N^{*}_{X/\mathbb{P}^{N}}(1))\subset\mathbb{P}^{N}\times(\mathbb{P}^{N})^{*}$. We denote by

[TABLE]

the canonical projections. Let us put $\displaystyle Z=q(p^{-1}(X^{*}(1)))$. Assume that $\displaystyle\dim Z=1$. Then, there exists an irreducible component of $\displaystyle Z$ which is included in $\displaystyle(H\cap X)_{tan}$ for all $\displaystyle H^{\perp}\in X^{*}$. Since $\displaystyle\dim X^{*}(1)=N-3$ and $\displaystyle N\geq 5$, this would imply that there is a $\displaystyle\mathbb{P}^{2}$ which is tangent to $\displaystyle X$ along $\displaystyle Z$. By Zak’s Theorem on Tangency, this is impossible. As a consequence, we have:

[TABLE]

so that the family of tangency curves in $\displaystyle X$ of hyperplanes parametrized by $\displaystyle X^{*}(1)$ is at least $\displaystyle 1$-dimensional.

Assume that this family is at least $\displaystyle 2$-dimensional. For any $\displaystyle H^{\perp}\in X^{*}$, we denote by $\displaystyle X_{H}$ the tangency locus of $\displaystyle H$ with $\displaystyle X$. Since the family $\displaystyle\{X_{H}\}_{H^{\perp}\in X^{*}(1)}$ is assumed to be at least $\displaystyle 2$-dimensional, we find that for any $\displaystyle x,y\in X$, there exists a hyperplane $\displaystyle H^{\perp}\in X^{*}(1)$ such that $\displaystyle x$ and $\displaystyle y$ lie on the curve $\displaystyle X_{H}$. Since $\displaystyle\dim X^{*}(1)=N-3$, Theorem 2.1.4 implies that for generic $\displaystyle H^{\perp}\in X^{*}(1)$, the linear span of $\displaystyle(X_{H})_{red}$ is $\displaystyle 2$-dimensional, which proves that $\displaystyle(X_{H})_{red}$ is a plane curve. Assume that $\displaystyle\deg(X_{H})_{red}\geq 3$. As for any $\displaystyle x,y\in X$, there exists $\displaystyle H^{\perp}\in X^{*}(1)$ such that $\displaystyle x$ and $\displaystyle y$ lie on the curve $\displaystyle(X_{H})_{red}$, this shows that every bisecant to $\displaystyle X$ is at least a trisecant. This is impossible by the trisecant lemma. We deduce that for generic $\displaystyle H^{\perp}\in X^{*}(1)$, the curve $\displaystyle(X_{H})_{red}$ is a plane conic. Note that for generic $\displaystyle H^{\perp}\in X^{*}(1)$, the conic $\displaystyle(X_{H})_{red}$ can not be singular as $\displaystyle X$ does not contain a two dimensional family of lines (otherwise $\displaystyle X$ would be a $\displaystyle\mathbb{P}^{2}$ linearly embedded in $\displaystyle\mathbb{P}^{5}$). We infer that $\displaystyle X$ is covered by an at least two-dimensional family of conics, the generic member of this family of conics being smooth. We conclude that $\displaystyle X$ is the Veronese surface in $\displaystyle\mathbb{P}^{5}$.

$\displaystyle\blacktriangleleft$

It would be interesting to know if one can get a classification result as in Theorem 3.1.1 when $\displaystyle\dim X^{*}(1)=N-3$ and the family of tangency loci along $\displaystyle X$ of hyperplanes parametrized by $\displaystyle X^{*}(1)$ is exactly $\displaystyle 1$-dimensional. By Theorem 2.2.3, this is equivalent to the following problem:

Problem 3.1.2

Classify all smooth surfaces $\displaystyle X\subset\mathbb{P}^{N}$ with $\displaystyle N\geq 5$ such that there is a $\displaystyle 1$-dimensional family of $\displaystyle\mathbb{P}^{3}$ whose tangency loci with $\displaystyle X$ are curves.

This classification must necessarily take into account ruled surfaces as many of them do satisfy the hypothesis of the problem.

3.2 A Bertini-type Theorem for osculating planes of space curves

In this section we will focus on a Bertini-type result for osculating hyperplanes that can be obtained as a consequence of Theorem 2.2.3.

Theorem 3.2.1

Let $\displaystyle C\subset\mathbb{P}^{3}$ be a smooth space curve. A generic osculating plane to $\displaystyle C$ is not tangent to $\displaystyle C$ outside its osculating locus.

Proof :

$\displaystyle\blacktriangleright\;$ We proceed by absurd. Denote by $\displaystyle C_{osc}\subset(\mathbb{P}^{3})^{*}$ the curve of osculating planes to $\displaystyle C$. We know (see [Pie83]) that for generic $\displaystyle H^{\perp}\in C_{osc}$, the plane $\displaystyle H$ osculates $\displaystyle X$ to first order at exactly one point, which we denote by $\displaystyle c_{H}$. Assume that for generic $\displaystyle H^{\perp}\in C_{osc}$, the plane $\displaystyle H$ is tangent to $\displaystyle C$ outside $\displaystyle c_{H}$. We denote by $\displaystyle C_{H}^{extra}$ the reduced tangency locus of $\displaystyle H$ with $\displaystyle C$ located outside $\displaystyle c_{H}$. Since $\displaystyle\dim C_{osc}=1$, Theorem 2.2.3 implies that for generic $\displaystyle H^{\perp}\in C_{osc}$, we have:

[TABLE]

where $\displaystyle T_{(H\cap C)_{tan},c_{H}}$ is the tangent space to the tangency locus of $\displaystyle H$ along $\displaystyle C$ at $\displaystyle c_{H}$. As $\displaystyle H$ is osculating $\displaystyle C$ at $\displaystyle c_{H}$ to the first order, we know that $\displaystyle T_{(H\cap C)_{tan},c_{H}}=T_{C,c_{H}}$. We deduce that the tangent $\displaystyle T_{C,c_{H}}$ cuts $\displaystyle C$ outside of $\displaystyle c_{H}$. This is true for generic $\displaystyle H^{\perp}\in X_{osc}$, so that for all $\displaystyle c\in C$, the tangent $\displaystyle T_{C,c}$ cuts again $\displaystyle C$ outside of $\displaystyle c$. Since $\displaystyle C$ is smooth, we get a contradiction with Kaji’s result on tangentially degenerate curve [Kaj86]. $\displaystyle\blacktriangleleft$

It is well known (and often used when dealing with plane projections of space curves) that a generic curve in $\displaystyle\mathbb{P}^{3}$ has only finitely many osculating planes which are tangent to the curve outside the osculating locus [BRF92], [GH94] (chapter 2, section 5), [Wal08]. To the best of my knowledge, it was not known that all smooth curves in $\displaystyle\mathbb{P}^{3}$ are generic, when it comes to the dimension of the family of doubly tangent osculating hyperplanes.

Remark 3.2.2

The reflexivity Theorem for osculating varieties **[Pie83]** shows that any space curve in $\displaystyle\mathbb{P}^{3}$ has a $\displaystyle 1$-dimensional family of bitangent osculating hyperplanes if and only if the generic tangent line to the osculating curve is at least a trisecant. On the other hand, the proof of the main result of **[Kaj86]** only works for curves which are mildly singular (see **[BP11]** for some improvements of Kaji’s result). It would be interesting to know if one can get a direct proof of Theorem 3.2.1 which would work for singular curves. This would imply in particular that any space curve is tangentially non-degenerate. 2. 2.

One expects that similar results to Theorem 3.2.1 hold for higher dimensional varieties. Namely, under appropriate hypotheses on the dimension and codimension of $\displaystyle X\subset\mathbb{P}^{N}$, it is not inconsiderate to believe that a generic osculating hyperplane is not tangent to $\displaystyle X$ outside of its osculating locus.

3.3 The algebraic boundary of an affine compact real variety

Let $\displaystyle X_{\mathbb{R}}\subset\mathbb{R}^{N}$ be an affine compact real variety. We denote by $\displaystyle P:=\mathrm{Conv}(X_{\mathbb{R}})$ the closed convex hull of $\displaystyle X$ in $\displaystyle\mathbb{R}^{N}$ and by $\displaystyle\partial_{a}P$ the Zariski closure in $\displaystyle\mathbb{P}^{N}_{\mathbb{C}}$ of the boundary of $\displaystyle\mathrm{Conv}(X_{\mathbb{R}})$. The complex projective hypersurface $\displaystyle\partial_{a}P$ is called the algebraic boundary of $\displaystyle X_{\mathbb{R}}$ and has become recently an object of active study for both algebraic geometry and optimization theory (see [RS11, RS12, BHO*+*12, ORSV15, Sin15] for instance).

In [RS11], a result describing the algebraic boundary of an affine compact real variety in terms of duals of some singular strata of the projective dual of its projective closure was proved under a restricting hypothesis: namely that only finitely many hyperplanes are tangent to the complex projective closure of the given variety at infinitely many points. Using Theorem 2.1.4, we are able to get rid of the restricting hypothesis.

Theorem 3.3.1

Let $\displaystyle X_{\mathbb{R}}\subset\mathbb{R}^{N}$ be a smooth and compact real algebraic variety that affinely spans $\displaystyle\mathbb{R}^{N}$. We denote by $\displaystyle X\subset\mathbb{P}^{N}_{\mathbb{C}}$ the Zariski closure of $\displaystyle X_{\mathbb{R}}$ in $\displaystyle\mathbb{P}^{N}_{\mathbb{C}}$. Then we have:

[TABLE]

where $\displaystyle r(X)$ is the minimal integer $\displaystyle k$ such that $\displaystyle(k+1)$-th secant variety of $\displaystyle X$ is of dimension at least $\displaystyle N-1$. In particular, every irreducible components of $\displaystyle\partial_{a}P$ is a component of $\displaystyle(\tilde{X}^{*}_{k})^{*}$ for some $\displaystyle k$.

Proof :

$\displaystyle\blacktriangleright\;$ The proof is exactly the same as the one that appears in [RS11] (proof of Theorem 1.1), except that our Theorem 2.1.4 improves and supersedes their Lemma 3.1. $\displaystyle\blacktriangleleft$

In [RS12, BHO*+*12, ORSV15], the study of the algebraic boundary of $\displaystyle X_{\mathbb{R}}\subset\mathbb{R}^{N}$ is done in connection with spectrahedral properties of $\displaystyle\mathrm{Conv}(X_{\mathbb{R}})$. More precisely, it is shown that if $\displaystyle\mathrm{Conv}(X_{\mathbb{R}})$ is a spectrahedron, then the algebraic boundary of $\displaystyle X$ has a rich geometry similar to that of classical determinantal hypersurfaces. Nevertheless, we are still lacking a precise description of the affine real compact varieties $\displaystyle X_{\mathbb{R}}\subset\mathbb{R}^{N}$ such that $\displaystyle\mathrm{Conv}(X_{\mathbb{R}})$ is spectrahedral. As part of a general work on spectrahedral representations of convex hulls of semi-algebraic set, Helton and Nie [HN09] conjectured that any semi-algebraic convex sets in $\displaystyle\mathbb{R}^{N}$ is a spectrahedral shadow. Scheiderer recently exhibited many counter-examples to this conjecture [Sch18]. In particular, Scheiderer proves the following:

Proposition 3.3.2** ([Sch18], proof of corollary 4.25)**

The closed convex hull of the Veronese embedding $\displaystyle v_{d}(\mathbb{R}^{n})\subset\mathbb{R}^{N+1}$ is not a spectrahedral shadow for $\displaystyle n\geq 4$ and $\displaystyle d\geq 4$.

It is difficult not to see that the numerical conditions that appear in the above statement are exactly the same that ensure that Conjecture 1.0.2 fails for the Veronese embedding $\displaystyle v_{d}(\mathbb{P}^{n-1})\subset\mathbb{P}^{N}$ (see Example 1.0.3 in the introduction). One could then wonder if the properties of being a spectrahedral shadow and verifying Conjecture 1.0.2 are related. Namely, one can ask:

Question 3.3.3

Let $\displaystyle X_{\mathbb{R}}\subset\mathbb{R}^{N}$ be an affine variety, we denote by $\displaystyle X$ the Zariski closure of $\displaystyle X_{\mathbb{R}}$ in $\displaystyle\mathbb{P}^{N}_{\mathbb{C}}$. Assume that $\displaystyle\mathrm{Conv}(X_{\mathbb{R}})$ is a spectrahedral shadow. Let $\displaystyle r$ be a positive integer, is it true that:

[TABLE]

where $\displaystyle X^{*}_{r}:=\{H^{\perp}\in X^{*},\,\textrm{such that}\,\dim\langle(H\cap X)_{tan}\rangle\geq r\}$?

A positive answer to this question would provide a quite effective criterion to decide if the convex hull of an affine real variety is a spectrahedral shadow.