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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 $\widetilde{Q}\cup E_p\cup E_q$ ?
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    $\begingroup$ Can't you do a computation in local charts? $\endgroup$ – Sasha May 22 '14 at 19:01
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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.


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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