Hilbert scheme

In algebraic geometry , the parameterized Hilbert scheme , the sub-schemes of projective space .

Hilbert functor

The Hilbert functor orders for a polynomial ${\displaystyle P}$

${\displaystyle h_{P}\colon \left\{Schemes\right\}^{op}\to \left\{Sets\right\}}$

to each scheme the set of over flat sub -schemas whose fibers have over points from Hilbert's polynomial . ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle {\mathcal {X}}\subset P_{B}^{n}}$ ${\displaystyle B}$ ${\displaystyle P}$

Hilbert scheme

For a polynomial Hilbert scheme is that the functor performing scheme. is therefore the point functor of . ${\displaystyle P}$ ${\displaystyle {\mathcal {H}}_{P}}$ ${\displaystyle h_{P}}$ ${\displaystyle h_{P}}$ ${\displaystyle {\mathcal {H}}_{P}}$

The uniqueness of follows from Yoneda's lemma , while existence is the result of a difficult construction. ${\displaystyle {\mathcal {H}}_{P}}$

Examples

Graßmann schemes

→ Main article : Graßmann scheme

The Graßmann scheme parameterizes the sub -schemes of degree 1 and dimension in for . But these are exactly the schemes whose Hilbert polynomial is. The Graßmann scheme is the Hilbert scheme for this polynomial. ${\displaystyle G(d,n)}$ ${\displaystyle d}$ ${\displaystyle P_{S}^{n}}$ ${\displaystyle S=\mathbb {Z} \left[x_{0},\ldots ,x_{n}\right]}$ ${\displaystyle H_{x}(m)=\left({\begin{array}{c}m+d\\d\end{array}}\right)}$

Hilbert scheme for hypersurfaces

The hypersurfaces of degree im are parameterized by the projective space of the vector space of the homogeneous polynomials of degree in variables. This projective space is the Hilbert scheme of the hypersurfaces of degree . ${\displaystyle d}$ ${\displaystyle P_{K}^{n}}$ ${\displaystyle d}$ ${\displaystyle n+1}$ ${\displaystyle d}$

Individual evidence

1. ^ David Mumford : Lectures on curves on an algebraic surface . Annals of Mathematical Studies 59, Princeton University Press 1966.
2. ↑ Janós Kollár : Rational curves on algebraic varieties . Results of Mathematics, 3rd Volume 32, Springer-Verlag Berlin 1996.