Shadows of Varieties Embedded in Projective Spaces

Abstract

When two varieties X, $X^{\prime }$ embedded in a projective space have the same image, i.e., the same shadow, are they projected from the same points? We prove that two general points of projections are sufficient to identify X. For one point of projection, there are many very different shadows with very different degrees. We give the geometric properties of some of them. These shadows are birational to the variety in which they are a shadow. We compute the minimum degree of all such shadows. For most smooth varieties $X\subset {\mathbb{P}}^r$, $r\ge 3$, it is the integer $deg\left(X\right)-1$.

1. Introduction

Take an integral variety $X\subset {\mathbb{P}}^r$, $r\ge 3$, of dimension $n>0$. For each $p\in {\mathbb{P}}^r$ let $C_p\left(X\right)$ be the cone with vertex p and X as a basis, i.e., the closure in ${\mathbb{P}}^r$ of the union of all lines spanned by p and a different point of X. The cone $C_p\left(X\right)$ is the shadow that X makes from p. If $Y\subset {\mathbb{P}}^r$ is an integral positive-dimensional variety $Y\subset {\mathbb{P}}^r$, we say that $(Y,p)$ is a shadow of X or of $(X,p)$ if $C_p\left(X\right)=C_p\left(Y\right)$. The pair $(X,p)$ has many very different shadows $(Y,p)$. Even if $dimY=n$, they may have arbitrarily large degrees. In some cases, we describe all of them (Theorems 8 and 9 and Remark 5).

In Section 3, we prove that if $(Y,p)$ is a shadow, it is not a shadow of $(X,q)$ for a general $q\in {\mathbb{P}}^r$, i.e., we prove the following result.

Theorem 1.

Fix $p\in {\mathbb{P}}^r$ and integral and non-degenerate varieties $X\subset {\mathbb{P}}^r$, $Y\subset {\mathbb{P}}^r$ such that $X\ne Y$ and $(Y,p)$ is a shadow of $(X,p)$. Set $n:=dimX$ and assume $n\le r-2$. Then, there is $q\in {\mathbb{P}}^r\setminus X$ such that $(Y,q)$ is not a shadow of $(X,q)$.

Then, in the same section, we prove that by using two general points we may avoid all shadows of X, i.e., we prove the following result.

Theorem 2.

Fix an integral and non-degenerate variety $X\subset {\mathbb{P}}^r$. Set $n:=dimX$. Assume $n\le r-2$. Then, there are $q_1,q_2\in {\mathbb{P}}^r\setminus X$ such that there is no integral variety Y with $(Y,q_1)$ and $(Y,q_2)$ shadows of X.

In Section 3, we define and briefly study longer shadows of X, i.e., shadows defined by the cone $C_V\left(X\right)$ with a positive-dimensional vertex V and X as its base. See Theorem 7 for the extension of Theorem 2 to longer shadows. These theorems are easy to prove because we fix X and then take projections from general points. This is not the subject of the very important topic (multiview) in which the points of projections, i.e., the pin-holes, are fixed, while the source, here X, moves; r is often 3 and the sets X are often finite unions of lines or conics ([1,2,3,4,5,6]). They determined (using some pin-holes) which point of X arrived, not if the arrived is X or another variety of Y. In the last part of Section 6, we discuss how the two topics interact.

Let us point it out again. In this paper, we see how good linear projections are in distinguishing different embedded varieties. We only use algebro-geometric tools as even the definitions do not involve a metric, except briefly in Remark 11. Metrics are crucial to measure the errors. Over the complex numbers, projective spaces have a natural metric, the Fubini–Study metric, but metrics would not help for our theorems.

To describe the possible shadows of X for all $p\in {\mathbb{P}}^r$, we introduce the following notation.

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Assume $dimX\le r-2$. Let $\delta \left(X\right)$ denote the minimum degree of all shadows of X (Theorem 6). Now, we see why $\delta \left(X\right)\le deg\left(X\right)-1$. For each $p\in X$, let ${\mu }_p\left(X\right)$ denote the minimum of the degrees of the connected component of $L\cap X$ with p as its reduction for the lines $L\subset {\mathbb{P}}^r$ containing p and a different point of X. The integer ${\mu }_p\left(X\right)$ is sometimes called the multiplicity of X at p. The integer ${\mu }_p\left(X\right)$ may be different from the integers obtained from other definitions of multiplicity (Remark 4). Since ${\mu }_p\left(X\right)\ge 1$ for all $p\in X$, the inequality $\delta \left(X\right)\le deg\left(X\right)-1$ follows from (1) below.

To understand more and discuss the case of arbitrary points of projections, we need to discuss the simple geometry of linear projections. The non-trivial theorems on non-birational outer projection were proved by A. Calabri and C. Ciliberto [7] and extended to inner projections (the ones giving the integer $k_1(X,p)$) and positive characteristics by K. Furukawa in [8].

To explain the degrees and the birational geometry of the shadows of X, we need to explain the following sets ${\mathfrak{S}}^0\left(X\right)$ and ${\mathfrak{S}}^1\left(X\right)$ and the integers $k_0(X,p)$ and $k_1(X,p)$. Let ${\mathfrak{S}}^0\left(X\right)$ denote the set of all $p\in {\mathbb{P}}^r\setminus X$ such that the morphism ${\mathsf{\ell }}_{p|X}:X\to {\mathsf{\ell }}_p\left(X\right)$ is not birational onto its image. For all $p\in {\mathfrak{S}}^0\left(X\right)$ set $k_0(X,p):=deg\left({\mathsf{\ell }}_{p|X}\right)$. We have $deg\left(k_0(X,p)\right)\ge 2$ because $p\in {\mathfrak{S}}^0\left(X\right)$. If $p\in {\mathbb{P}}^r\setminus {\mathfrak{S}}^0\left(X\right)$, we write $k_0(X,p):=1$. Let ${\mathfrak{S}}^1\left(X\right)$ denote the set of all $p\in X$ such that X is not a cone with vertex containing p and the map ${\mathsf{\ell }}_{p|X\setminus \{p\}}$ has degree $\ge 2$ onto its image. We call this degree $k_1(X,p)$. If X is a cone with vertex p, we set $k_1(X,p):=0$. If $p\in X$, and p is not a cone with vertex containing p and $p\notin {\mathfrak{S}}^1\left(X\right)$, we set $k_1(X,p):=1$. We have

$$k_1(X,p)deg\left(C_p\left(X\right)\right)=(deg\left(X\right)-{\mu }_p\left(X\right))/k_1(X,p)$$

(1)

if X is not a cone with vertex containing p.

As a sample of the results, we prove in this paper, as we put in the introduction, the following two points and briefly discuss them.

Theorem 3.

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Set $n:=dimX$ and assume $n\le r-2$. Let $(Y,p)$ be a shadow of X such that $dimY=n$. If $p\notin (X\cup Y\cup {\mathfrak{S}}^0\left(X\right)\cup {\mathfrak{S}}^0\left(Y\right))$, then $deg\left(X\right)=deg\left(Y\right)$ and X and Y are birational.

There are many $(Y,p)$ as in Theorem 3. In some cases, one can hope to obtain a full classification, but we leave this topic to the interested readers.

Theorem 4.

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Set $n:=dimX$ and assume $n\le r-2$. Fix $p\notin X$.

(a) 

Let $(Y,p)$ be a shadow of X such that $dimY=n$ and $p\notin Y$. Then, $deg\left(Y\right)=kdeg\left(X\right)$ for some positive integer k.

(b) 

All positive integers k appear for some X and $k=1$ if and only if $p\notin {\mathfrak{S}}^0\left(X\right)$.

There are many $(Y,p)$ as in Theorem 4. The examples in Theorem 4 are given as the intersection of $C_p\left(X\right)$ and a sufficiently general degree k hypersurface. Hence, for $k=1$ they are degenerate. In Theorems 8 and 9, we discuss when they are the only ones. The integer k occurring in Theorem 4 is the integer $k_0(Y,p)$.

If we can handle the geometry of the linear projection from a given point, we may exclude several types of shadows. See Proposition 3.

It is natural to restrict the places at which we allow the linear projections. We prove the following result.

Theorem 5.

Let X, Y and W be integral and non-degenerate subvarieties of ${\mathbb{P}}^r$. Assume that we have $dimX\le dimY\le min\{r-2,dimW\}$. If $dimW=dimY$, assume $W\notin \{X,Y\}$ and that W is not a linear space. If ${\mathsf{\ell }}_w\left(X\right)={\mathsf{\ell }}_w\left(Y\right)$ for a general $w\in W$, then $X=Y$.

In Section 6, we discuss how projections of points help to reconstruct X, even if only approximate solutions are known.

We conclude with the Section 7.

We work over an algebraically closed field $\mathbb{K}$ of characteristic 0. We leave to the interested reader the extension of this paper, i.e., the positive characteristic case (avoiding some of the use of the theorem of Bertini and always quoting [8] instead of [7]).

We thank the referees for useful suggestions.

2. Preliminary Results

For any set $S\subset {\mathbb{P}}^r$, let $\langle S\rangle$ denote its linear span, i.e., the intersection of all hyperplanes of ${\mathbb{P}}^r$ containing S with the convention $\langle S\rangle ={\mathbb{P}}^r$ if S is not contained in a hyperplane. Note that $\langle S\rangle$ is the minimal linear subspace of ${\mathbb{P}}^r$ containing S. We use the same notation for all projections.

Remark 1.

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Set $n:=dimX$. Take $p\in X$. Note that $p\in {\mathfrak{S}}^1\left(X\right)$ if and only if either X is a cone with vertex containing p (case $X=C_p\left(X\right)$, i.e., $dim({\mathsf{\ell }}_p(X\setminus \left\{p\right\})=n-1$), or a general secant line of X containing p is a multisecant line of X (case $dim{\mathsf{\ell }}_p(X\setminus \left\{p\right\})=n$). Now, assume $X\ne C_p\left(X\right)$ and call $X^{\prime }$ the closure of ${\mathsf{\ell }}_p(X\setminus \left\{p\right\})$. Set $d:=deg\left(X\right)$. Let e be the number of points of $L\cap (X\setminus \{p\left\}\right)$ for a general line $L\subset {\mathbb{P}}^r$ containing p. We have $d=edeg\left(X^{\prime }\right)+{\mu }_p\left(X\right)$.

Remark 2.

If $X={\mathbb{P}}^r$, the possible solutions $(Y,p)$ with $Y\ne X$ are given by the hypersurfaces of ${\mathbb{P}}^r$ with p any point not in the vertex of Y. Assume $r=n+1$. The possible solutions $(Y,p)\ne (X,p)$ are given by $({\mathbb{P}}^r,p)$ with p not in the vertex of X and the hypersurfaces Y with p neither in the vertex of Y nor in the vertex of X or, if X is a cone with vertex containing p, some $(r-2)$ dimensional varieties as in the following Remark 3.

Remark 3.

Take $(Y,p)$ such that $C_p\left(X\right)=C_p\left(Y\right)$ and $dimX\ne dimY$. Recall that either we have $C_p\left(X\right)=X$ or that $dim{C}_p\left(X\right)=dimX+1$ and that either we have $C_p\left(Y\right)=Y$ or we have $dim{C}_p\left(Y\right)=dimY+1$. Thus, $dimY=dimX-1$ and p is contained in the vertex of X or $dimY=dimX+1$ and Y is any $(dimX-1)$-dimensional integral subvariety of $C_p\left(X\right)$ which is not a cone with vertex containing p. Given X, it is obvious how to test its vertex (if any) and hence the possible pairs $(Y,p)$ for the case are $dimY=dimX-1$. It is even easier to describe the set of pairs $(Y,p)$ such that $C_p\left(Y\right)=C_p\left(X\right)$ and $dimY=dimX+1$. Indeed, take any $p\in {\mathbb{P}}^r$ such that X is not a cone with vertex containing p and set $Y:=C_p\left(X\right)$.

Remark 4.

Let $X⊊{\mathbb{P}}^r$ be an integral variety and $p\in X$. Set $n:=dimX$. Let V be a general $(r-n)$-dimensional linear subspace containing p. By the theorem of Bertini, the algebraic set $X\cap V$ is smooth outside p ([9], Cor. III.10.9, [10], I.6.3). The scheme–theoretic intersection $X\cap V$ has degree $deg\left(X\right)$. Set $m:=deg\left(X\right)-s+1$, where s is the number of connected components of $X\cap V$. If X is smooth at p, then $m={\mu }_p\left(X\right)=1$. Now, assume that X is singular and let $T_{p}X$ be the Zariski tangent space of X at p ([11], pp. 85–88). We get $m\ge {\mu }_p\left(X\right)$ with strict inequality if $dim{T}_{p}X\ge n+2$. If $dim{T}_{p}X=n+1$, then all the reasonable definitions of multiplicity of X at p coincide and $m={\mu }_p\left(X\right)$.

Remark 5.

Let $X\subset {\mathbb{P}}^r$ be an integral n-dimensional variety. We have ${\mathfrak{S}}^0\left(X\right)=\varnothing$, unless there is a huge number of planes containing 4 points of X. We have no such point of X if either X is a c-Veronese embedding of X for some $c\ge 3$, i.e., a linearly normal embedding of X with ${\mathcal{O}}_X\left(1\right)={\mathcal{R}}^{\otimes c}$, or a general linear projection of it and $r\gg dimX$.

We have ${\mathfrak{S}}^1\left(X\right)=\varnothing$ if X does not have a huge number of planes containing at least five points of X and a huge number of trisecant lines. To avoid the latter the case ${\mathcal{O}}_X\left(1\right)={\mathcal{R}}^{\otimes 2}$ is sufficient.

If C is a very general smooth curve of genus $g\ge 6$, then ${\mathfrak{S}}^0\left(X\right)={\mathfrak{S}}^1\left(X\right)=\varnothing$ for every embedding $X\subset {\mathbb{P}}^r$ of C (note that we must have $r>2$).

Proposition 1.

Take $p\in {\mathbb{P}}^r$ such that X is not a cone with vertex p and let u be the degree of the closure $X_1$ of ${\mathsf{\ell }}_p(X\setminus \left\{p\right\})$. Take $(Y,p)$ such that $(Y,p)$ is a shadow of X and $dimY=dimX$. Then, $deg\left(Y\right)\ge u$ and $deg\left(Y\right)=u$ implies $p\notin Y$, $p\notin {\mathfrak{S}}^1\left(Y\right)$ and Y birational to $X_1$.

Proof. 

By assumption $Y\subset C_p\left(Y\right)=C_p\left(X\right)$, $deg\left(C_p\left(X\right)\right)=u$ and ${\mathsf{\ell }}_p$ induces a rational and dominant map $v:X⤏X_1$. Hence, $deg\left(Y\right)\ge ku+c$, where $k=deg\left(v\right)$ is a positive integer and $c={\mu }_p\left(Y\right)$ (which is a non-negative integer). We have ${\mu }_p\left(Y\right)=0$ if and only if $p\notin Y$. □

Proposition 2.

Take $p\in X$ such that X is a cone with vertex containing p. Set $d:=deg\left(X\right)$. Let $(Y,p)$ be a shadow of X. If $dimY=dimX$, then $Y=X$. If $dimY\ne dimX$, then $dimY=dimX-1$ and $deg\left(Y\right)={\mu }_p\left(Y\right)+kd$ with k positive integer k the degree of the rational map induced by the linear projection from p. We have $k>1$ if and only if $p\in {\mathfrak{S}}^1\left(Y\right)$.

Proof. 

Since $(Y,p)$ is a shadow of Y, we have $C_p\left(Y\right)=C_p\left(X\right)$. Since

$$dim{C}_p\left(Y\right)-1\le dimY\le dim{C}_p\left(X\right)$$

and $C_p\left(X\right)=X$, $dimY=dimX$ if and only if $Y=X$. Now, assume $dimY<dimX$ and hence $dimY=dim{C}_p\left(Y\right)$. Let $Y_1$ be the closure of ${\mathsf{\ell }}_p(Y\setminus \left\{p\right\})$ in ${\mathbb{P}}^{r-1}$. We have $deg\left(C_p\left(Y\right)\right)=deg\left(X\right)$ and $deg\left(C_p\left(Y\right)\right)=deg\left(Y_1\right)$. The linear projection from p shows that $deg\left(Y\right)={\mu }_p\left(Y\right)+kd$. □

As a summary of our observations and results, we obtain the following result.

Theorem 6.

Assume $n\le r-2$. Set $d:=deg\left(X\right)$. The integer $\delta \left(X\right)$ is the minimum of the following two integers $a_1$ and $a_2$: If ${\mathfrak{S}}^0\left(X\right)=\varnothing$, then set $a_1:=d$. If ${\mathfrak{S}}^0\left(X\right)\ne \varnothing$, then let $a_1$ be the minimum of all integers $d/k_0(X,p)$, $p\in {\mathfrak{S}}^0\left(X\right)$. The integer $a_2$ is the minimum of all the integers $deg\left(X\right)/k_1(X,p)$, $p\in X$, p not a vertex of X, with the convention $k_1(X,p)=1$ if $p\in X\setminus {\mathfrak{S}}^1\left(X\right)$.

Proof. 

This is a consequence of Proposition 1 and the case of X as a cone considered in Proposition 2. □

3. Longer Shadows and Proofs of Theorems 1 and 2

In this section, we briefly describe the case in which we take cones with respect to positive-dimensional linear subspaces of ${\mathbb{P}}^r$.

For all integers $0\le m\le r$ let $G(m+1,r+1)$, denote the Grassmannian of the m-dimensional linear subspaces of ${\mathbb{P}}^r$. The set $G(m+1,r+1)$ is a connected projective manifold of dimension $(r-m)(m+1)$. For all $V\in G(m+1,r+1)$ and all integral varieties X, let $C_V\left(X\right)\subseteq {\mathbb{P}}^r$ denote the closure of the union of all spaces $\langle V\cup \left\{o\right\}\rangle$ with $o\in X$. Note that $dimX+m+1\ge dim{C}_V\left(X\right)\ge min\{m,dimX\}$, $C_V\left(X\right)=V$ if and only if $X\subset V$, and $C_V\left(X\right)=X$ if and only if X is a cone with vertex containing V. For any $V\in G(m+1,r+1)$ and any integral variety Y, we say that $(Y,V)$ is a shadow of X or that it is a shadow of $(X,V)$ if $C_V\left(X\right)=C_V\left(Y\right)$.

Remark 6.

If $V\in G(m+1,r+1)$ and $(Y,V)$ is a shadow of X, then

$$dimX-m-1\le dimY\le dimX+m=1.$$

Moreover, $dimY=dimX+m+1$ only if $Y=C_V\left(X\right)$ and $dimY=dimX-m-1$ only if X is a cone with vertex containing V.

Lemma 1.

Fix integers $m\ge 0$, $s>0$, $N\ge 2m+2$, a finite set $S\subset {\mathbb{P}}^N$ such that $\#S=s$ and $V\in G(m+1,N+1)$. Set $E:={\cup }_{o\in S}\langle \left\{o\right\}\cup V\rangle$. Take a general $W\in G(m+1,N+1)$ and set $F:={\cup }_{o\in S}\langle \left\{o\right\}\cup W\rangle$. Then, $E\cap F=S$.

Proof. 

If $s=1$, then the lemma is obvious because $V\cap W=S$ by the assumption $N\ge 2m+2$. Thus, we may assume $s\ge 2$. In this case, we just use the distributive law. □

Lemma 2.

Let $X\subset {\mathbb{P}}^r$ be an integral and n-dimensional variety. Fix an integer $m\ge 0$ such that $r\ge 2m+n+2$ and take a general $(V,W)\in G(m+1,r+1)\times G(m+1,n+1)$. Then, $dim(C_V\left(X\right)\cap C_W\left(X\right)\setminus X)\le n-1$.

Proof. 

Let $U\subset {\mathbb{P}}^r$ be a general $(r-n)$-dimensional linear subspace. By the Theorem of Bertini, the set $S:=U\cap X$ is formed by $s:=deg\left(X\right)$ distinct points. To prove the lemma it is sufficient to show that $U\cap C_V\left(X\right)\cap C_W\left(X\right)=S$. This is even true by Lemma 1 if we take V and W with the additional restriction that they are contained in U. □

Proof of Theorem 1.

First, assume $dimY=n+1$, i.e., $Y=C_p\left(X\right)$ and X is not a cone with vertex p. In this case, it is sufficient to take $q\in {\mathbb{P}}^r\setminus Y$ such that X is not a cone with q contained in its vertex. One can also use Lemma 2.

Second, assume $dimY=n-1$. We have $X=C_p\left(Y\right)$. Hence, in this case it is sufficient to take q outside the vertex of X.

Third, assume $dimY=n$. Since $X\ne Y$, $X\cap Y\ne Y$. Fix $a\in Y\setminus X\cap Y$. Since $n\le r-2$, $C_a\left(X\right)\ne {\mathbb{P}}^r$. Take $q\in {\mathbb{P}}^r\setminus (C_a\left(X\right)\cup Y)$. Since $q\notin X$, $dim{C}_q\left(X\right)=n+1$. Assume $C_q\left(X\right)=C_q\left(Y\right)$. Since $q\notin (X\cup Y)$, there is a line L containing $\{a,q\}$ and intersecting X, contradicting the definition of $C_q\left(Y\right)$. □

Proof of Theorem 2.

We first handle the n-dimensional shadows of X.

Fix a general $q\in {\mathbb{P}}^r$. Since $r>n$, $q\notin (X\cup {\mathfrak{S}}^0\left(X\right))$ and $dim{C}_q\left(X\right)=n+1$. Take a point $o\in {\mathbb{P}}^r\setminus C_q\left(X\right)$. By assumption the line $\langle \{o,q\}\rangle$ does not meet X. Hence, $C_o\left(X\right)\ne C_q\left(X\right)$. Thus, $C_o\left(X\right)\cap C_q\left(X\right)$ contains only finitely many n-dimensional irreducible components, one of them being X. Apply Theorem 1 to each of these n-dimensional irreducible components different from X, if any.

Now, we handle the $(n+1)$-dimensional shadows of X. They are of the form $C_p\left(X\right)$ for some p which is not a vertex of X. It is sufficient to take $q\in {\mathbb{P}}^r\setminus C_p\left(X\right)$, so that $C_q\left(X\right)\ne C_p\left(X\right)$ and hence $dim{C}_q\left(C_p\left(X\right)\right)=n+2$.

Now, we handle the $(n-1)$-dimensional shadows of X. They exist if and only if X is a cone and the point p must be one of the vertices of X. To avoid them it is sufficient to use q such that $dim{C}_q\left(X\right)=n+1$. □

Theorem 7.

Fix positive integers r, n, m such that $r\ge 2m+2+n$. Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate n-dimensional variety. For a general $(V,W)\in G{(m+1,r+1)}^2$ there is no n-dimensional variety Y such that $(Y,V)$ is a shadow of $(X,V)$ and $(Y,W)$ is a shadow of $(X,W)$.

Proof. 

Since $r\ge n+m+1$ and $(V,W)$ are general, we have

$$dim{C}_V\left(X\right)=dim{C}_W\left(X\right)=n+m+1.$$

Apply Lemma 2. □

4. The Geometry of the Shadows

Remark 7.

Take a positive integer k, an integral and non-degenerate variety $X\subset {\mathbb{P}}^r$ and a point $p\notin (X\cup {\mathfrak{S}}^0\left(X\right))$. There are many shadows $(Y,p)$ of $(X,p)$ such that $dimX=dimY$, $p\in Y$ and ${\mathsf{\ell }}_p$ induce a degree k rational map $Y⤏{\mathsf{\ell }}_p\left(X\right)$. These examples may be constructed as in the proof of Theorem 8 by twisting ${\mathcal{O}}_U\oplus {\mathcal{O}}_U\left(1\right)$, $U:={\mathsf{\ell }}_p\left(X\right)$, by a very positive line bundle on U. We always have $deg\left(Y\right)>kdeg\left(X\right)$. If $dimX\ge 2$ we are unable to construct Y which are smooth, even if X is smooth.

Theorem 8.

Let $X\subset {\mathbb{P}}^r$ be an integral, normal and non-degenerate variety. Set $n:=dimX$ and assume $n\le r-2$. Fix $p\notin (X\cup {\mathfrak{S}}^0\left(X\right))$.

(a) If X is linearly normal, then all shadows $(Y,p)$ with $dimY=n$, $p\notin Y$ and $p\notin {\mathfrak{S}}^0\left(Y\right)$ are the complete intersection of $C_p\left(X\right)$ with a hyperplane.

(b) Assume that X is not linearly normal. Then, there is a shadow $(Y,p)$ of X such that $dimY=n$, $p\notin (Y\cup {\mathfrak{S}}^0\left(Y\right))$, $deg\left(Y\right)=deg\left(X\right)$, $Y\subset C_p\left(X\right)$, Y spans ${\mathbb{P}}^r$ and X is isomorphic to the normalization of Y.

Proof. 

Take a shadow $(Y,p)$ with $p\notin Y$ and $p\notin {\mathfrak{S}}^0\left(Y\right)$.

Since $p\notin X$, $U:={\mathsf{\ell }}_p\left(X\right)$ is an n-dimensional variety and $\mu :={\mathsf{\ell }}_{p|X}:X\to U$ is a proper morphism. Since $p\notin {\mathfrak{S}}^0\left(X\right)$, $\mu$ is a birational morphism and $deg\left(U\right)=deg\left(X\right)$. Since $p\notin X$, no line through p is contained in X. Thus, $\mu$ is finite. Since $\mu$ is birational and finite and X is normal, $\mu$ is the normalization map. Let $v_1:\tilde{\mathbb{P}}\to {\mathbb{P}}^r$ denote the blowing up of p and let $C_1\left(X\right)\subset \tilde{\mathbb{P}}$ denote the strict transform of the variety $C_p\left(X\right)$. There is a morphism $v_2:C_1\left(X\right)\to U$ which makes $C_1\left(X\right)$ the projectivization of the vector bundle ${\mathcal{O}}_U\oplus {\mathcal{O}}_U(-1)$. Let $v_3:C_2\left(X\right)\to X$ denote the pull-back of $v_2$ by the normalization map. The morphism $v_3$ is the projectivization of the vector bundle ${\mathcal{O}}_X\oplus {\mathcal{O}}_X(-1)$. Note that the composition of $C_2\left(X\right)\to C_1\left(X\right)$ and $v_1$ sends each integral n-dimensional variety $Y^{\prime }$ of $C_2\left(X\right)$ mapped by $v_3$ to X and is not contained in the pull-back of $v_1^{-1}\left(p\right)\cap C_1\left(X\right)$ to an n-dimensional integral subvariety of $C_p\left(X\right)$. Parts (a) and (b) come from the linear system $|{\mathcal{O}}_X\left(1\right)\oplus {\mathcal{O}}_X|$ and the structure of sections of $v_3$ not intersecting the pull-back of $v_1^{-1}\left(p\right)\cap C_1\left(X\right)$ (see [9], Prop. II.7.12). □

Theorem 9.

Let $X\subset {\mathbb{P}}^r$ be an integral, normal and non-degenerate variety. Set $n:=dimX$ and assume $n\le r-2$. Fix $p\notin (X\cup {\mathfrak{S}}^0\left(X\right))$ and an integer $k>1$. There is a shadow $(Y,p)$ of X such that $p\notin Y$, $dimY=n$, $deg\left(Y\right)=kdeg\left(X\right)$ and Y is not the intersection of $C_p\left(X\right)$ with a degree k hypersurface if and only if the restriction map $H^0({\mathbb{P}}^r,{\mathcal{O}}_{{\mathbb{P}}^r}\left(k\right))\to H^0(X,{\mathcal{O}}_X\left(k\right))$ is not surjective.

(a) If X is linearly normal, then all shadows $(Y,p)$ with $dimY=n$, $p\notin Y$ and $p\notin {\mathfrak{S}}^0\left(Y\right)$ are the complete intersection of $C_p\left(X\right)$ with a hyperplane.

(b) Assume that X is not linearly normal. Then, there is a shadow $(Y,p)$ of X such that $dimY=n$, $p\notin (Y\cup {\mathfrak{S}}^0\left(Y\right))$, $deg\left(Y\right)=deg\left(X\right)$, $Y\subset C_p\left(X\right)$, Y spans ${\mathbb{P}}^r$ and X is isomorphic to the normalization of Y.

Proof. 

The proof of Theorem 8 works with no modification for $k>1$, quoting again [9], Prop. II.7.12. □

Proof of Theorem 3.

Since $p\notin (X\cup {\mathfrak{S}}^0\left(X\right))$, the cone $C_p\left(X\right)$ has degree $deg\left(X\right)$ and ${\mathsf{\ell }}_p\left(X\right)$ is birational to X. For the same reason $deg\left(C_p\left(Y\right)\right)=deg\left(Y\right)$ and ${\mathsf{\ell }}_p\left(Y\right)$ is birational to Y. Since ${\mathsf{\ell }}_p\left(X\right)={\mathsf{\ell }}_p\left(Y\right)$, we conclude. □

Proof of Theorem 4.

Since $p\notin Y$, ${\mathsf{\ell }}_{p|Y}$ is a finite map. Let k be its degree. Since we have ${\mathsf{\ell }}_p\left(Y\right)={\mathsf{\ell }}_p\left(X\right)$ and $deg\left({\mathsf{\ell }}_p\left(X\right)\right)=deg\left(X\right)$, we obtain part (a).

We claim that to prove part (b), it is sufficient to take as Y the intersection of $C_p\left(X\right)$ with a general degree k hypersurface T. The generality of T implies $p\notin T$. The generality of T and the theorem of Bertini ([10]. Th. I.6.3) imply that $C_p\left(X\right)\cap T$ is a hypersurface of $C_p\left(X\right)$ of degree $kdeg\left(X\right)$ intersecting the general line contained in $C_p\left(X\right)$ and containing p in k distinct points. Since $p\notin (T\cup X)$, we get ${\mathsf{\ell }}_p\left(Y\right)={\mathsf{\ell }}_p\left(X\right)$. □

Proof of Theorem 5.

Since $dimW\ge max\{dimX,dimY\}$ and $W\notin \{X,Y\}$, $w\notin (X\cup Y)$ for a general $w\in W$. Since W is not a linear space if $dimY=dimW$ and Y is not a hypersurface, [7], Th. 2.6, this implies that ${\mathsf{\ell }}_w\left(X\right)$ is birational to X and ${\mathsf{\ell }}_w\left(Y\right)$ is birational to Y. Hence, $dimX=dimY$. Assume $X\ne Y$. Consider a general $a\in X\setminus X\cap Y$. Let $C_a\left(Y\right)$ be the cone with vertex a and Y as a base (it may have a positive-dimensional vertex even if Y is not a cone). We have $dim{C}_a\left(Y\right)\le dimX+1<r$. For a general $w\in W$ there is $u\in Y$ such that ${\mathsf{\ell }}_w\left(u\right)={\mathsf{\ell }}_w\left(u\right)$. Note that $w\in C_a\left(Y\right)$. Hence, $W\subseteq C_a\left(Y\right)$. For a general $u\in X\setminus \left\{a\right\}$, we have $C_u\left(Y\right)\ne C_a\left(Y\right)$, because X is non-degenerate and the vertex of $C_u\left(Y\right)$ is a linear subspace of ${\mathbb{P}}^r$. Since $W\subseteq C_a\left(Y\right)\cap C_u\left(Y\right)$, $dimW\le dimX$; a contradiction. □

Recall that if we can handle the geometry of the linear projection from a given point, then we may exclude several types of shadows.

In the following proposition we only need to handle one linear projection, ${\mathsf{\ell }}_p$, if we restrict X. Note also that ${\mathfrak{S}}^0\left(X\right)=\varnothing$ is almost always true (Remark 5) and for a specific $p\in {\mathbb{P}}^N$ testing if $p\in ({\mathfrak{S}}^0\left(X\right)\cup {\mathfrak{S}}^1\left(X\right))$ we only need ${\mathsf{\ell }}_p$ to do the test.

Proposition 3.

Fix an integral and non-degenerate variety $X\subset {\mathbb{P}}^r$ such that the identity is the only birational automorphism of X. Take $p\in {\mathbb{P}}^r\setminus (X\cup {\mathfrak{S}}^0\left(X\right))$ and let $(Y,p)$ a shadow of $(X,p)$ with $p\notin ({\mathfrak{S}}^0\left(Y\right)\cup Y)$. If $q\notin (X\cup {\mathfrak{S}}^0\left(X\right))$, then $(Y,q)$ is not a shadow of $(X,q)$.

Proof. 

Assume that $(Y,q)$ is a shadow of $(X,q)$. Set $d:=deg\left(X\right)$. Since $p\notin ({\mathfrak{S}}^0\left(Y\right)\cup Y)$, we have $deg\left(Y\right)=d$. Note that ${\mathsf{\ell }}_p$ induces a birational morphism $u_p$ between X and Y. Hence, the identity map is the only birational automorphism of Y and ${\mu }_p$ is the unique birational morphism between X and Y. Since $q\notin (X\cup {\mathfrak{S}}^0\left(X\right))$, we have $deg\left(C_q\left(X\right)\right)=d$. Since $C_q\left(X\right)=C_q\left(Y\right)$, we get $q\notin Y$ and hence $deg\left({\mathsf{\ell }}_q\left(Y\right)\right)=d/k_0(Y,q)$. Hence, we have $k_0(Y,q)=1$. Therefore, ${\mathsf{\ell }}_q$ induces a birational map $u_q$ between X and Y. Hence, $u_q=u_p$, i.e., for a general $a\in X$ we have ${\mathsf{\ell }}_q\left(b\right)=a$, where b is the unique point of $Y\cap \langle \{a,p\}\rangle$. Thus, $\langle \{q,p\}\rangle =\langle \{a,b\}\rangle$. For a general $a\in X$ we get $q=p$; a contradiction. □

5. Necessary Conditions for Shadows

In this section, we collect some criteria which imply that two subvarieties of ${\mathbb{P}}^r$ are not shadows of each other.

Proposition 4.

Let X, Y, be subvarieties such that $X\ne Y$ and there is $p\in {\mathbb{P}}^r$ with $(Y,p)$ a shadow of X. Then, $X\cap Y\ne \varnothing$ and $dimX\cap Y\ge max\{dimX-1,dimY-1\}$.

Proof. 

By assumption $X\ne Y$, $X\subseteq C_p\left(X\right)$, $Y\subseteq C_p\left(Y\right)$ and $C_p\left(X\right)=C_p\left(Y\right)$. Hence, the proposition is true if either $X=C_p\left(X\right)$ or $Y=C_p\left(Y\right)$. Thus, we may assume $n:=dimX=dimY=dim{C}_p\left(X\right)-1$. Since $X\ne Y$, $dimX\cap Y\le n-1$.

(a) Assume $n=1$. In this case, it is sufficient to prove that $X\cap Y\ne \varnothing$. If $p\in X\cap Y$, then we are done. Hence, we may assume that p is contained in at most one among X and Y. Let $D\subset {\mathbb{P}}^{r-1}$ denote the closure of ${\mathsf{\ell }}_p(X\setminus \left\{p\right\})$. The integral curve C is projectively equivalent to a general hyperplane section of $C_p\left(X\right)$, say $D=C_p\left(X\right)\cap H$ with H a general hyperplane. We will see D as a subset of H and hence as a subset of $C_p\left(X\right)$. Let $\nu :C\to D$ denote the normalization map. Let ${\nu }_1:C_p{\left(X\right)}^{\prime }\to C_p\left(X\right)$ denote the pull-back morphism induced by $\nu$. Call $p^{\prime }\subset C_p{\left(X\right)}^{\prime }$ the counterimage of p. The rational map ${\mathsf{\ell }}_p$ induces a smooth surjection $\mathsf{\ell }:C_p{\left(X\right)}^{\prime }\setminus \left\{p^{\prime }\right\}\to C$ with fibers isomorphic to the affine line. Let S denote the blowing up of $C_p{\left(X\right)}^{\prime }$ at p and let $\pi :S\to C$ denote the rational map induced by ${\mathsf{\ell }}_p$. The variety S is a smooth surface and the rational map $\pi$ is a smooth ${\mathbb{P}}^1$ bundle in the sense of [9]. Let $X^{\prime }$ and $Y^{\prime }$ be the strict transform of X and Y in S. It is sufficient to prove that $X^{\prime }\cap Y^{\prime }\ne \varnothing$. Hence, it is sufficient to prove that $X^{\prime }·Y^{\prime }>0$ (intersection number of the irreducible curves $X^{\prime }$ and $Y^{\prime }$). Since ${\mathcal{O}}_C\left(1\right)$ is very ample, inside the ruled surface S the curve J mapped to $p^{\prime }$ is the unique irreducible curve with negative self-intersection (the integer $deg\left(D\right)$), while the fibers of $\pi$ are the only irreducible curves of S with zero as their self-intersection. The counterimage of a general hyperplane section of $C_p\left(X\right)$ and a fiber of $\pi$ are a basis of the Neron–Severi group of S. If $p\in Y$, then $Y^{\prime }$ is ample and hence $X^{\prime }·Y^{\prime }>0$. Thus, we may assume $p\notin (X\cup Y)$. In this case, $c{X}^{\prime }+J$ is very ample for $c\gg 0$. Since $J·Y^{\prime }=0$ and $c{X}^{\prime }+J$ is very ample for $c\gg 0$, we have $X^{\prime }·Y^{\prime }>0$.

(b) Now, assume $n>1$. It is sufficient to take the intersection of X and Y with a general codimension $n-1$ linear subspace $V\subset {\mathbb{P}}^r$. Note that $p\notin V$ and that $X\cap V$ and $Y\cap V$ are integral by the theorem of Bertini (they are degenerate, but in the case $n=1$, we never use or assume that X or Y is non-degenerate). □

Recall that $(Y,p)$ is a shadow of $(X,p)$ if and only if ${\mathsf{\ell }}_p\left(X\right)={\mathsf{\ell }}_p\left(Y\right)$. Thus, we make the following two observations (Proposition 5 and Remark 8).

Proposition 5.

Fix $p\in {\mathbb{P}}^r$ and let ${\mathsf{\ell }}_p:{\mathbb{P}}^r\setminus \left\{p\right\}\to {\mathbb{P}}^{r-1}$ denote the linear projection from p. Fix a hyperplane $H\subset {\mathbb{P}}^r$ such that $p\notin H$ and see ${\mathsf{\ell }}_p$ as the composition of the surjective morphism ${\mathsf{\ell }}_p:{\mathbb{P}}^r\setminus \left\{p\right\}\to H$ and the inclusion $H\subset {\mathbb{P}}^r$, so that the target of ${\mathsf{\ell }}_p$ is contained in ${\mathbb{P}}^r$.

(a) 

The rational map ${\mathsf{\ell }}_p$ is a flat limit of a family of elements of $PGL(r+1)$.

(b) 

Let $X\subset {\mathbb{P}}^r$ be an integral variety such that $p\notin X$ and ${\mathsf{\ell }}_{p|X}X\to {\mathsf{\ell }}_p\left(X\right)$ is an isomorphism. Then, ${\mathsf{\ell }}_p\left(X\right)$ is a flat limit of a family of varieties projectively equivalent to X.

Proof. 

Fix a system $x_0,\dots ,x_n$ of homogeneous coordinates such that $H=\{x_n=0\}$ and $p=[0:\cdots :0:1]$. For all $c\in \mathbb{K}\setminus \left\{0\right\}$ let $h_c$ be the element of $\mathrm{Aut}\left({\mathbb{P}}^r\right)$ defined by the formula $h_c\left([a_0:\cdots :a_{n-1}:a_n]\right)=[a_0:\cdots :a_{n-1},c{a}_n]$. Note that $h_p=h_0$ (ref. [9], Example III.9.4.3). For part (b) use part (a) and that ${\mathsf{\ell }}_p\left(X\right)$ and X have the same Hilbert polynomial because the linear projection $h_p$ induces the isomorphism between X and $h_p\left(X\right)$ (ref. [9], Theorem III.9.9). □

For any integral variety $X\subset {\mathbb{P}}^r$, let ${\sigma }_2\left(X\right)$ denote the secant variety of X, i.e., the closure in ${\mathbb{P}}^r$ of the union of all lines spanned by two points of X.

Remark 8.

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate n-dimensional variety such that ${\sigma }_2\left(X\right)\ne {\mathbb{P}}^r$ and fix $p\in {\mathbb{P}}^r\setminus {\sigma }_2\left(X\right)$. Set $\mu :={\mathsf{\ell }}_p\left(X\right)$. Since $p\notin X$, $\mu :X\to {\mathsf{\ell }}_p\left(X\right)$ is a morphism. Since $p\notin {\sigma }_2\left(X\right)$, μ is injective. Hence, if X is normal the morphism $\mu :X\to {\mathsf{\ell }}_p\left(X\right)$ is the normalization map. For each $q\in X_{\mathrm{reg}}$ the tangent space $T_{q}X$ of X at q is contained in ${\sigma }_2\left(X\right)$. Since $p\notin {\sigma }_2\left(X\right)$, μ has non-zero differential at p. Hence, μ is an isomorphism if X is smooth. Assume that X is smooth. We understand that $X\cong {\mathsf{\ell }}_p\left(X\right)$ (isomorphism of varieties, not projective equivalence). By Proposition 5 the variety ${\mathsf{\ell }}_p\left(X\right)$ is a flat limit of a family of subvarieties projectively equivalent to X.

6. Fixed Pin-Holes and Reconstruction from the Projected Points

In the first part of this section, we give some remarks when instead of an irreducible variety X we have a finite union of irreducible varieties and we want to reconstruct all of them. Equivalently, we have a reducible variety $X\subset {\mathbb{P}}^r$. Let $X_1,\dots ,X_s$ be the irreducible components of X. We obtain some images and we try to reconstruct the union $X_1\cup \cdots \cup X_s$, but not the order of the components. If we have some well-posed projections sufficient to reconstruct each $X_i$, then we have won. By Theorem 2 we know that two sufficiently general linear projections are sufficient for all $X_1,\dots ,X_s$. Random linear projections usually work. However, we need to be able to choose the linear projections. Errors in the images may occur.

As in the applications to multiview, we take a receiving hyperplane M (the screen) and we only use projections from points of ${\mathbb{P}}^r\setminus M$ and the target is M, not an abstract $(r-1)$-dimensional projective space.

Remark 9.

Consider a linear projection of X from $p\in {\mathbb{P}}^r\setminus M$. If the image contains something of dimension $\le n-2$, then we know that it is wrong. If we see something, T, of dimension $n-1$ and we are sure that there was no error, then $C_p\left(T\right)$ was one of the irreducible components of X.

Example 1.

Consider a linear projection of X from $p\in {\mathbb{P}}^r\setminus M$, $r\ge 3$. Take $n=1$ and that in M we do not look at isolated points. If we obtain nothing else, then $X=L_1\cup \cdots \cup L_s$ with $L_1,\dots ,L_s$ lines containing p. Take $q\in {\mathbb{P}}^r\setminus M$ and look at one-dimensional images. They must be (if there are no errors) either s lines or $s-1$ lines. In the latter case one of the lines is $\langle \{p,q\}\rangle$, while the other ones may be reconstructed in the following way. Take a line $L\subset {\mathbb{P}}^r$ containing p and a point $q\notin L$. The set $T:={\mathsf{\ell }}_q\left(L\right)$ is a line and L is contained in the plane $C_q\left(T\right)$. Consider $o:={\mathsf{\ell }}_q\left(p\right)$. Since $p\notin M$, $p\ne o$. Hence, L is the line $\langle \{p,o\}\rangle$.

In the applications often each $X_i$ is very simple, say a line or a conic, and we want to reconstruct it from a fixed set of point of projections.

The first problem is that if we project a line from one of its points we only obtain a point when X is reducible. For the moment we fix a positive integer n, assuming that each connected component of X has dimension n, and only look at images by ${\mathsf{\ell }}_p$ of dimension n and take only the n-dimensional images.

Projection of Points

Now, we try to reconstruct X from some of its projected points. A key step is knowing something about X, e.g., that it is a line or a conic or it is cut out by quadrics.

Remark 10.

Fix $p,q,o\in {\mathbb{P}}^r\setminus M$ such that $dim\langle \{p,q,o\}\rangle =2$. Note that

$$\left\{o\right\}=\langle \{p,{\mathsf{\ell }}_p\left(o\right)\rangle \cap \langle \{q,{\mathsf{\ell }}_q\left(o\right)\rangle .$$

Hence, o is uniquely reconstructed from two linear projections, while a unique linear projection only restricts the place in which o must be searched.

For a line it is sufficient to have two of its points. For a smooth conic it is sufficient to have five of its points and these points must be coplanar and no three of them collinear.

If they are not coplanar, then there was an error. If they are coplanar, three of them are coplanar and there was no error, then it was sent a reducible conic.

Now, assume that X is cut out by quadric hypersurfaces. Call $H^0\left({\mathcal{O}}_{{\mathbb{P}}^r}\left(2\right)\right)$ the vector space of all degree 2 forms in $r+1$ variables. It has dimension $\left({\displaystyle \genfrac{}{}{0pt}{}{r+2}{2}}\right)$. We fix a homogeneous system of coordinates $x_0,\dots ,x_r$ of ${\mathbb{P}}^r$ and we take the coefficients of the monomials of the degree 2 forms as a basis of it. Suppose from the sent points (using two or more pin-holes) we obtained a set $S\subset {\mathbb{P}}^r$. Take $\widehat{S}\subset {\mathbb{K}}^{n+1}\setminus \left\{0\right\}$ with $\#\widehat{S}=\#S$. The evaluation of $H^0\left({\mathcal{O}}_{{\mathbb{P}}^r}\left(2\right)\right)$ at the points of $\widehat{S}$ gives a system of linear equations. We obtain a linear subspace of degree 2 forms. Call this system T. We have $X\subseteq {\{f=0\}}_{f\in T}$. If we are allowed to obtain enough sufficiently general points of X, we obtain the reconstruction of X. If $\mathbb{K}=\mathbb{C}$, X is defined over $\mathbb{R}$ and the set $X\left(\mathbb{R}\right)$ of real points of X is Zariski dense in the set $X\left(\mathbb{C}\right)$ of all complex-valued points of X, then it is sufficient to know a sufficient number of smooth real points of X.

We may use also forms of higher degrees, instead of quadrics, only we need larger linear systems and a large number of

Remark 11.

Take as a base field the real number or the complex number with the usual euclidean topology. In this case, we may use a metric inducing the euclidean topology. For instance, or ${\mathbb{P}}^r\left(\mathbb{C}\right)$ and hence on ${\mathbb{P}}^r\left(\mathbb{R}\right)$ we may use the Fubini–Study metric. In this set-up we may have approximate data and hence approximate solutions. We would like to have a good approximation of the variety X. Remark 10 cannot be used (the intersection of the approximate lines is empty). The set-up with homogeneous equations is better, but only if we knew that X is a complete intersection of forms of degrees $d_1,\dots ,d_{r-n}$. In this case, with enough points we recover X using equations. If we only have an a approximate finite set $S_{ϵ}$ with $S_{ϵ}$ giving the right number of conditions to forms of degree $d_1,\dots ,d_{r-n}$, then we obtain a complete intersection n-dimensional variety $X_{ϵ}$. The variety $X_{ϵ}$ is a good approximation of X if a priori we know that X is the complete intersection of forms of degree $d_1,\dots ,d_{r-n}$.

7. Discussion

We consider how to describe an embedded variety from its image by linear projections from points. We briefly mention that it is related to the geometry of multiview ([1]), but the flavor, tools and results are very different. All our tools come from Algebraic Geometry. In this set-up the main results on linear projection we use are due to A. Calabri, C. Ciliberto and K. Furukawa. We prove that two general points of projections are sufficient. For one point of projection there are many very different shadows (with very different degrees).

We give the geometric properties of some of them. For instance, they are birational to the variety of which they are a shadows. We compute the minimum degree of all such shadows. For most varieties X it is the integer $deg\left(X\right)-\mu$ with $\mu \ge 1$ and $\mu =1$ if X is smooth. Often it is easy to check the point of projections with shadows of degrees of at most $deg\left(X\right)-2$. Over the real or complex numbers, we discuss approximate solutions when our data are approximate. We discuss why many of our tools and proof would not work, while one may be used. We briefly consider longer shadows, i.e., linear projections from higher dimensional linear subspaces.