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pi_1
pi_1
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Let $k$ be an algebraically closed field of characteristic $0$. For a homogeneous ideal $I=(q_1,\dots, q_k)\subset k[x_0,\dots,x_n]$ generated by quadrics, is there a method to decide whether the generators of $I$ admit relations "of degree $1$" i.e. if there are linear polynomials $L_i\in k[x_0,\dots,x_n]_1$ such that $\sum_iL_ig_i=0$? Can one compute (in explicit cases) the dimension of such $k$-uples of linear polynomial?

For a homogeneous ideal $I=(q_1,\dots, q_k)\subset k[x_0,\dots,x_n]$ generated by quadrics, is there a method to decide whether the generators of $I$ admit relations "of degree $1$" i.e. if there are linear polynomials $L_i\in k[x_0,\dots,x_n]_1$ such that $\sum_iL_ig_i=0$? Can one compute (in explicit cases) the dimension of such $k$-uples of linear polynomial?

Let $k$ be an algebraically closed field of characteristic $0$. For a homogeneous ideal $I=(q_1,\dots, q_k)\subset k[x_0,\dots,x_n]$ generated by quadrics, is there a method to decide whether the generators of $I$ admit relations "of degree $1$" i.e. if there are linear polynomials $L_i\in k[x_0,\dots,x_n]_1$ such that $\sum_iL_ig_i=0$? Can one compute (in explicit cases) the dimension of such $k$-uples of linear polynomial?

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pi_1
pi_1
  • 1.5k
  • 8
  • 13

Decide whether there are "linear" relations between quadrics

For a homogeneous ideal $I=(q_1,\dots, q_k)\subset k[x_0,\dots,x_n]$ generated by quadrics, is there a method to decide whether the generators of $I$ admit relations "of degree $1$" i.e. if there are linear polynomials $L_i\in k[x_0,\dots,x_n]_1$ such that $\sum_iL_ig_i=0$? Can one compute (in explicit cases) the dimension of such $k$-uples of linear polynomial?