Dimension of the space of cubic polynomials over $\mathbb{P}^5$ which vanish on the Veronese surface.

Question

How should one go about computing the dimension of the space of cubic polynomials over $\mathbb{P}^5$ which vanish on $\mathbb{P}^2$, where $\mathbb{P}^2$ sits inside $\mathbb{P}^5$ via Veronese embedding. (Here $\mathbb{P}^k$ is complex projective space). I am more interested in elementary approaches.

Answer 1

$\newcommand{\PP}{{\mathbb P}}$ $\newcommand{\Ohol}{{\mathcal O}}$ $\newcommand{\QQ}{{\mathbb Q}}$ $\newcommand{\sheaf}[1]{{\mathcal #1}}$

You can do the following calculation with Macaulay2. The idea is the following: One considers the sequence

$$0 \to \sheaf{I} \to \Ohol_{\PP^5} \to v_*\Ohol_{\PP^2} \to 0$$

where $v:\PP^2 \to \PP^5$ is the Veronese-imbedding. One applies $\bigoplus_d\Gamma(\PP^5,- \otimes \Ohol_{\PP^5}(d))$ and gets

$$0 \to I \to S \to R$$

with $S=\QQ[t_0,\ldots,t_5]$ and $R=\QQ[x,y,z]$ where $v:S \to R$ is given by $t_0 \mapsto x^2,\ldots,t_5 \mapsto z^2$.

Now the vanishing cubic polynomials are $I_3 = \Gamma(\PP^5,\sheaf{I}(3))$ which can be selected by the command basis(3,id1) where id1 is the ideal $I$ in Macaulay2 (line i13 below). One counts $28$ columns in the result matrix, which is therefore the dimension of the space of cubic polynomials vanishing on the veronese-surface. To double check, one computes basis(3,S/id1) (line i15), which gives $28$ too. As the cubic polynomials have dimension $\binom{5 + 3}{3} = 56$, our calculation must be right.

i1 : R=QQ[x,y,z]

o1 = R

o1 : PolynomialRing

i2 : S=QQ[t_0..t_5]

o2 = S

o2 : PolynomialRing

i3 : describe S

o3 = QQ[t , t , t , t , t , t , Degrees => {6:1}, Heft => {1}, MonomialOrder =>
         0   1   2   3   4   5                                                 

     -----------------------------------------------------------------------------------------------
     {MonomialSize => 32}, DegreeRank => 1]
     {GRevLex => {6:1}  }
     {Position => Up    }

i5 : basis(2,R)

o5 = | x2 xy xz y2 yz z2 |

             1       6
o5 : Matrix R  <--- R

i6 : vmat = oo

o6 = | x2 xy xz y2 yz z2 |

             1       6
o6 : Matrix R  <--- R

i7 : phi=map(R,S,vmat)

               2             2        2
o7 = map(R,S,{x , x*y, x*z, y , y*z, z })

o7 : RingMap R <--- S


i12 : ker phi

              2                                    2                       2
o12 = ideal (t  - t t , t t  - t t , t t  - t t , t  - t t , t t  - t t , t  - t t )
              4    3 5   2 4    1 5   2 3    1 4   2    0 5   1 2    0 4   1    0 3

o12 : Ideal of S

i13 : basis(3, ker phi)

o13 = {2} | t_3 t_4 t_5 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  
      {2} | 0   0   0   t_3 t_4 t_5 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  
      {2} | 0   0   0   0   0   0   t_2 t_3 t_4 t_5 0   0   0   0   0   0   0   0   0   0   0   0  
      {2} | 0   0   0   0   0   0   0   0   0   0   t_0 t_1 t_2 t_3 t_4 t_5 0   0   0   0   0   0  
      {2} | 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   t_0 t_1 t_2 t_3 t_4 t_5
      {2} | 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  
      ----------------------------------------------------------------------------------------------
      0   0   0   0   0   0   |
      0   0   0   0   0   0   |
      0   0   0   0   0   0   |
      0   0   0   0   0   0   |
      0   0   0   0   0   0   |
      t_0 t_1 t_2 t_3 t_4 t_5 |

o13 : Matrix

i14 : S1=S/(ker phi)

o14 = S1

o14 : QuotientRing

i15 : basis(3,S1)

o15 = | t_0^3 t_0^2t_1 t_0^2t_2 t_0^2t_3 t_0^2t_4 t_0^2t_5 t_0t_1t_3 t_0t_1t_4 t_0t_1t_5 t_0t_2t_5
      ----------------------------------------------------------------------------------------------
      t_0t_3^2 t_0t_3t_4 t_0t_3t_5 t_0t_4t_5 t_0t_5^2 t_1t_3^2 t_1t_3t_4 t_1t_3t_5 t_1t_4t_5
      ----------------------------------------------------------------------------------------------
      t_1t_5^2 t_2t_5^2 t_3^3 t_3^2t_4 t_3^2t_5 t_3t_4t_5 t_3t_5^2 t_4t_5^2 t_5^3 |

               1        28
o15 : Matrix S1  <--- S1