pencil of quadrics consisting of singular quadrics A pencil $l$ of quadrics in $\mathbb{P}^4_{<x_0,\cdots,x_4>}$ consists of singular quadrics only if:
(1) quadrics in $l$ have a common singular point; or
(2) quadrics in $l$ contain a common plane; or
(3) restricted to a common hyperplane, the quadrics in $l$ are singular along a line;
Furthermore, if $l$ satisfies (3) but not (1) or (2), then up to projectively $l=(x_0x_2-x_1^2,x_2x_4-x_3^2)$. 
How can I prove? And I want to know the case when we replace $\mathbb{P}^4$ by $\mathbb{P}^3$(or $\mathbb{P}^5$). 
There are many references about pencils of quadrics whose generic elements are nonsingular and I can't find a reference about pencil of quadrics which consists of singular quadrics. Can You give any references?
 A: Let me just sketch a straightforward approach and leave some details to fill in.
Let $\mathbb{P}^4=\mathbb{P}(V)$, where $\dim V = 5$. The space of all quadrics is nothing but $S=\mathbb{P}^{14}=\mathbb{P}(S^2V^*)$.
Saying that $\ell$ consists of singular quadrics means that $\ell\subset \Delta$, where $\Delta$ is the degree $5$ hypersurface in $S$ consisting of degenerate quadrics. Consider the stratification given by corank: $S\supset\Delta=\Delta_1\supset\Delta_2\supset\Delta_3\supset\Delta_4$. In particular, $\Delta_4$ is nothing but the image of the second Veronese embedding $v_2(\mathbb{P}(V))$. A good exercise is to check that $\mathrm{Sing}\ \Delta_i = \Delta_{i+1}$.
Let us denote by $s(Q)$ the singular locus of $Q)$. It's not hard to check that the projective tangent space $T_Q\Delta$ to $Q\in\Delta\setminus\Delta_2$ is nothing but the space of quadrics passing through the point $s(Q)$ and that the tangent cone to $Q\in\Delta_2\setminus\Delta_3$ is the set of quadrics tangent to $l=s(Q)$.
Remark I. There are no lines $\ell\subset \Delta_4=v_2(\mathbb{P}(V))$. In particular, at least one quadric in the pencil has corank $\geq 3$.
Remark II. If at least two quadrics in the pencil have corank $\geq 3$, we are in case (1): intersect the kernels of the corresponding correlation maps.
Remark III. If two quadrics contain some plane, any quadric from the pencil contains this plane.
Remark IV. If two quadrics have a common singular point, all the pencil has this singular point.
Assume that $\ell\subset\Delta_2$. If $\ell$ contains a quadric of corank 4, we are done by Remark III. If $\ell$ contains a quadric $Q_3$ of corank 3, either we are in case (1), or $s(Q_2)$ and $s(Q_3)$ are complementary and $\ell$ contains a smooth quadric.
Assume that $\ell\cap (\Delta\setminus \Delta_2)\neq \emptyset$. Pick three quadrics from the pencil of corank 1. If the three vertices are in general position, then all of the quadrics contain the plane passing through them. Otherwise, vertices form a line $l\in\mathbb{P}^4$ that is contained in all the quadrics. If the latter holds and $\ell\subset\Delta\setminus \Delta_2$, consider the relative scheme of planes in quadrics containing $l$. This would give a nonramified degree two covering of $\ell=\mathbb{P}^1$. Such a covering would be trivial, and the corresponding component curves should intersect in the variety of planes in $\mathbb{P}(V)$ containing $l$, which is $\mathbb{P}^2$. Thus, two quadrics from the pencil contain the same plane and we are in case (2).
You are left with the cases when $\ell\subset\Delta_2\setminus\Delta_3$ or $\ell\subset\Delta_1\setminus\Delta_3$, all of them contain a line $l$, and $\ell\cap\Delta_2\neq\emptyset$. I'll let you finish the argument (case (3) gives a nice hint).
