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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 $4\times 4$ 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.

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One of my favourite sets of examples, stolen from Miles Reid, is the determination of rings $R=\oplus_m H^0(X, mD)$ 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 $4\times 4$ 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.

...

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 $2\times 2$ 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.

...

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 $4\times 4$ 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.

...

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 $P^1$.

$n=3$: get $E\subset P^2$, a general cubic.

$n=4$: 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.

...

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 $2\times 2$ 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.

...