Let $Q\subset\mathbb{P}^n$ be the quadric hypersurface defined by
$$x\_0^2+x\_1^2+...+x\_k^2 =0.$$
If $2\leq k\leq n-1$ then $Q$ is irreducible and $Sing(Q)$ is a linear space of dimension $n-k-1$.


* If $n = 3$, $k=2$, then $Q\subset\mathbb{P}^3$ is a quadric cone. If $\pi:X\rightarrow\mathbb{P}^3$ is the blow-up of the vertex $p$ of $Q$ with exceptional divisor $E\_p$, then the strict transform $\widetilde{Q}$ of $Q$ is smooth. Furthermore the divisor $\widetilde{Q}\cup E\_p$ in $X$ is simple normal crossing.
* Now, let us consider the case $n=4$, $k = 2$. Therefore $Q\subset\mathbb{P}^4$ is a cone of dimension $3$ over a smooth plane conic with vertex along a line $L$. Let us take two points $p,q\in L$, and let $\pi:X\rightarrow\mathbb{P}^4$ be the blow-up of $p,q$. Finally, let $\widetilde{Q}$ be the strict transform of $Q$, and let $E\_p,E\_q$ be the exceptional divisors. What can we say about the singularities of $\widetilde{Q}$ and $\widetilde{Q}\cup E\_p\cup E\_q$ ?



Singular irreducible quadrics
In the case $n=4$, $k=2$, one can perform an explicit computation in charts of the blow-up. Consider the quadric $Q$ defined by
$$\{x\_1^2+x\_2^2+x\_3^2=0\}\subset\mathbb{A}^4.$$
The singular line is given by $L={x\_1=x\_2=x\_3 = 0}$. Let us consider the points $p = (0,0,0,0)$ and $q = (0,0,0,1)$ on $L$. We can look at a local chart of the blow-up $Bl\_{p,q}\mathbb{A}^4$ in $\mathbb{A}^4\times\mathbb{A}^3\times\mathbb{A}^3\cong\mathbb{A}^{10}$. We will see that the strict transform $\widetilde{Q}$ of $Q$ is still singular along the strict transform $\widetilde{L}$ of $L$. Therefore $\widetilde{L}$ is a double curve for $\widetilde{Q}$. You can see this from the following MacAulay2 script. In higher dimension I am pretty sure that things go this way as well.




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Macaulay2, version 1.6
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone


i1 : A10 = QQ[x\_1,x\_2,x\_3,x\_4,y\_1,y\_2,y\_3,z\_1,z\_2,z\_3]


o1 = A10


o1 : PolynomialRing


i2 : J = ideal(x\_1\*y\_2-x\_2\*y\_1,x\_1\*y\_3-x\_3\*y\_1,x\_1-x\_4\*y\_1,x\_2\*y\_3-x\_3\*y\_2,x\_2-x\_4\*y\_2,x\_3-x\_4\*y\_3,x\_1^2+x\_2^2+x\_3^2,x\_1-z\_1\*x\_4-z\_1,x\_2-x\_4\*z\_2+z\_2,x\_3-x\_4\*z\_3+z\_3)



```
                                                                                             2    2    2

```

o2 = ideal (- x y + x y , - x y + x y , - x y + x , - x y + x y , - x y + x , - x y + x , x + x + x , - x z + x - z , - x z + x + z , -
 2 1 1 2 3 1 1 3 4 1 1 3 2 2 3 4 2 2 4 3 3 1 2 3 4 1 1 1 4 2 2 2 



```
 x z  + x  + z )
  4 3    3    3

```

o2 : Ideal of A10


i3 : L=primaryDecomposition J



```
                                                                                                                                      2    2  

```

o3 = {ideal (x z - x - z , x z - x - z , x z - x + z , x z + y z + x z - y z + x z - y z , x y - x , x y - x , x y - x y , y + y +
 4 3 3 3 4 2 2 2 4 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 4 3 3 4 2 2 3 2 2 3 1 2 



```
  2                                        2    2    2   2              2      2                  2       2                2       2        
 y , x y  - x , x y  - x y , x y  - x y , x  + x  + x , y z  - y y z , y z  + y z  + y y z  + 2y z  - 4z z  + y y z  + 2y z  - 4z z ), ideal
  3   4 1    1   3 1    1 3   2 1    1 2   1    2    3   3 2    2 3 3   2 1    3 1    1 2 2     1 2     1 2    1 3 3     1 3     1 3        

                              2                     2                            2                                                            
 (x  + z , x  + z , x  - z , z , z z , z z , x z , z , z z , y z  - y z , x z , z , y z  + y z , y z  + y z , x z , x y  + z , x y  + z , x y 
   3    3   2    2   1    1   3   2 3   1 3   4 3   2   1 2   3 2    2 3   4 2   1   3 1    1 3   2 1    1 2   4 1   4 3    3   4 2    2   4 1

        2
 - z , x )}
    1   4

```

o3 : List


i4 : A = L#0



```
                                                                                                                                     2    2  

```

o4 = ideal (x z - x - z , x z - x - z , x z - x + z , x z + y z + x z - y z + x z - y z , x y - x , x y - x , x y - x y , y + y +
 4 3 3 3 4 2 2 2 4 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 4 3 3 4 2 2 3 2 2 3 1 2 



```
  2                                        2    2    2   2              2      2                  2       2                2       2
 y , x y  - x , x y  - x y , x y  - x y , x  + x  + x , y z  - y y z , y z  + y z  + y y z  + 2y z  - 4z z  + y y z  + 2y z  - 4z z )
  3   4 1    1   3 1    1 3   2 1    1 2   1    2    3   3 2    2 3 3   2 1    3 1    1 2 2     1 2     1 2    1 3 3     1 3     1 3

```

o4 : Ideal of A10


i5 : R = A10/A


o5 = R


o5 : QuotientRing


i6 : X = Spec(R)


o6 = X


o6 : AffineVariety


i7 : dim(X)


o7 = 3


i8 : Z = singularLocus(X)


o8 = Z


o8 : AffineVariety


i9 : dim(Z)


o9 : 1




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