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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 1

Decomposition of the given vanishing locus

Subtracting the two defining equations gives
$$
y^2+z^2=(y-iz)(y+iz)=0.
$$
Consequently the variety is
$$
V(y-iz,x^2-z^2-1)
\ \cup\
V(y+iz,x^2-z^2-1).
$$
Each member is irreducible. Indeed, after imposing its linear equation, its <coordinate ring> is
$$
\mathbb C[x,z]/(x^2-z^2-1).
$$
The invertible linear substitution
$$
u=x-z,\qquad v=x+z
$$
turns this into
$$
\mathbb C[u,v]/(uv-1)\cong\mathbb C[u,u^{-1}],
$$
which is an <integral domain>. The two curves are distinct and neither contains the other, so they are precisely the <irreducible components of two intersecting complex hyperbolas>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 25I / Decomposition of the given vanishing locus / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 25I / Decomposition of the given vanishing locus / Solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 25I / Decomposition of the given vanishing locus / Solution

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