Tight closure/x^4,y^4,xy^3/On P^1 and other curves/Example

We consider the ideal ${}I={\left(x^{4},y^{4},xy^{3}\right)}$ in ${}S=K[x,y]$ and in finite graded extensions ${}S\subseteq R$ (e.g. ${}R=K[x,y,z]/(F)$, where ${}F$ is a homogeneous integral equation for ${}z$ over $K[x,y]$) and describe an algorithm to compute the tight closure ${}I^{*}$. The graded resolution of the ideal is

$0\longrightarrow S(-5)\oplus S(-7)\longrightarrow S(-4)\oplus S(-4)\oplus S(-4){\stackrel {x^{4},y^{4},xy^{3}}{\longrightarrow }}S\longrightarrow S/I\longrightarrow 0,$

where the map on the left is given by sending the generators to

$(0,x,-y){\text{ and }}(y^{3},0,-x^{3}).$

On the projective line ${}{\mathbb {P} }_{K}^{1}$ this corresponds to

$0\longrightarrow {\mathcal {O}}(-5)\oplus {\mathcal {O}}(-7)=\operatorname {Syz} {\left(x^{4},y^{4},xy^{3}\right)}\longrightarrow {\mathcal {O}}(-4)\oplus {\mathcal {O}}(-4)\oplus {\mathcal {O}}(-4){\stackrel {x^{4},y^{4},xy^{3}}{\longrightarrow }}{\mathcal {O}}\longrightarrow 0.$

We may pull back this sequence along the finite morphism

$C=\operatorname {Proj} {\left(R\right)}\longrightarrow {\mathbb {P} }_{K}^{1}$

to obtain the corresponding exact sequence over the curve ${}C$ which can be used to compute the tight closure of the ideal in ${}R$. A homogeneous element ${}h\in R$ of degree ${}m$ yields a cohomology class in

${}H^{1}(C,\operatorname {Syz} {\left(x^{4},y^{4},xy^{3}\right)}(m))\cong H^{1}(C,{\mathcal {O}}_{C}(m-5))\oplus H^{1}(C,{\mathcal {O}}_{C}(m-7))\,,$

which can be easily computed using Čech cohomology. On ${}D_{+}(x)$, ${}h$ comes from ${}\left({\frac {h}{x^{4}}},0,0\right)$ and on ${}D_{+}(y)$ it comes from ${}\left(0,{\frac {h}{y^{4}}},0\right)$. Their difference, the syzygy

$\left({\frac {h}{x^{4}}},-{\frac {h}{y^{4}}},0\right)$

equals

$-{\frac {h}{xy^{4}}}\left(0,x,-y\right)+{\frac {h}{x^{4}y^{3}}}\left(y^{3},0,-x^{3}\right).$

Hence the components of this cohomology class are

$-{\frac {h}{xy^{4}}}\in H^{1}(C,{\mathcal {O}}_{C}(m-5)){\text{ and }}{\frac {h}{x^{4}y^{3}}}\in H^{1}(C,{\mathcal {O}}_{C}(m-7)).$

Therefore the issue whether ${}h$ belongs to the tight closure of ${}I$ depends on these two components, which both correspond to a parameter situation.

First of all, if ${}m\geq 7$, then both degrees are non-negative and therefore these classes are tightly ${}0$ by (the proof of) fact. If ${}m=6$, we only have to look at the second component inside ${}H^{1}(C,{\mathcal {O}}_{C}(-1))$. For the monomial ${}y^{3}z^{3}$ the second component is ${}0$ (independent of ${}F$), hence it belongs to the tight closure, though the first component need not be ${}0$. The monomial ${}x^{2}y^{2}z^{2}$ yields ${}{\frac {z^{2}}{x^{2}y}}$, which is not ${}0$ unless the equation has low degree. This class is (with some exceptions in small characteristics) not tightly ${}0$. Hence ${}x^{2}y^{2}z^{2}$ does not belong to the tight closure. For ${}m=5$, still only the second component is interesting, therefore ${}y^{3}z^{2}$ belongs to the tight closure, but ${}xy^{2}z^{2}$ not (under the same restrictions). For ${}m\leq 4$ both components lie in negative degree, so an element will belong to the tight closure only if it belongs to the ideal itself. For the element ${}y^{3}z$ the second component is ${}0$, but not the first component.