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One of my favourite sets of examples, stolen from Miles Reid, is the determination of rings $R=\oplus_n H^0(X, nD)$ for ample divisors $D$ on projective varieties $X$. A nice sequence, where a lot of the general features of the theory already show up, is to take $X=E$ an elliptic curve, and $D=nP$ for $P$ a point on $E$.

$n=1$: generators in degrees $1,2,3$, with a relation in degree 6 (by Riemann-Roch), leading to $E\subset P^2[1,2,3]$ (weighted projective space) a sextic hypersurface given by Weierstrass equation $z^2=y^3 + ax^3 y + bx^6$.

$n=2$: get $E\subset P^2[1,1,2]$, a double cover of $P^1$; $P\in E$ is one of the ramification points.

$n=3$: get $E\subset P^2$, a general cubic; $P\in E$ is an inflection point on the image.

$n=4$: get $E\subset P^3$, a general complete intersection of bidegree $(2,2)$.

$n=5$: get $E\subset P^4$, a non-complete intersection variety, equations are the $2\times 2$ 4\times 4$Pfaffians of a general$5\times 5$skew-symmetric matrix of linear forms, equivalently a linear section of$Gr(2,5)$in its Plucker embedding. ... 2 added 90 characters in body One of my favourite sets of examples, stolen from Miles Reid, is the determination of rings$R=\oplus_n H^0(X, nD)$for ample divisors$D$on projective varieties$X$. A nice sequence, where a lot of the general features of the theory already show up, is to take$X=E$an elliptic curve, and$D=nP$for$P$a point on$E$.$n=1$: generators in degrees$1,2,3$, with a relation in degree 6 (by Riemann-Roch), leading to$E\subset P^2[1,2,3]$(weighted projective space) a sextic hypersurface given by Weierstrass equation$z^2=y^3 + ax^3 y + bx^6$.$n=2$: get$E\subset P^2[1,1,2]$, a double cover of$P^1$.P^1$; $P\in E$ is one of the ramification points.

$n=3$: get $E\subset P^2$, a general cubic; $P\in E$ is an inflection point on the image.

$n=4$: get $E\subset P^3$, a general complete intersection of bidegree $(2,2)$.

$n=5$: get $E\subset P^4$, a non-complete intersection variety, equations are the $2\times 2$ Pfaffians of a general $5\times 5$ skew-symmetric matrix of linear forms, equivalently a linear section of $Gr(2,5)$ in its Plucker embedding.

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