# Tuesday, January 17

:::{.remark}
Recall $X = V(f_1,\cdots, f_m) = \spec R \subseteq \AA^n\slice k$ where $R\da \CC[x_1,\cdots, x_n]/\gens{f_1,\cdots, f_m}$ is smooth if $\Jac\ts{f_i} = \qty{\dd{f_i}{x_j}}$ has maximal rank $r = \codim_{\AA^n} X$ at *all* points $x\in X$, and we'll give a more intrinsic notion of smoothness which does not depend on the choice of equations $\ts{f_i}$.
Over $k=\CC$, if $X$ is smooth it is a complex manifold.
:::

:::{.example title="?"}
For $X\da V(x^2 + y^2 + 1)$, note $\nabla f = \tv{2x, 2y}$ has rank 1 everywhere except $0$, but since $0\not\in X$, in fact $X$ is smooth.
:::

:::{.definition title="Kähler differentials"}
Recall that $\Omega^1{R/k} \da \bigoplus R\dr/I$ where 
\[
I = \gens{ d(rs) = rds + sdr, d(cr) = cdr, d(r+s) = dr + ds \st r,s\in R, c\in k}
.\]
Note that $\Omega^1_{R/\CC}\in \rmod$, while $\Omega^1_{X/\CC}\in \oxmods$ is a sheaf.
:::

:::{.example title="?"}
Example: for $R = \CC[x,y]/\gens{x^2+y^2+1}$, we have $\Omega^1_{R/\CC} = Rdx + Rdy/I$.
Noting $x^2+y^2+1=0$ in $R$, we have 
\[
0 = d(x^2+y^2+1) = 2xdx + 2y dy 
.\]
:::

:::{.definition title="Smoothness"}
$X$ is **smooth** iff the rank of $\Omega^1_{X/\CC}$ at $p$ is $\dim X$ for every $p\in X$.
For $X$ a variety, point $p\in X$ correspond to $\mfm_p \in \mspec R$ and $\Omega^{1}_{R/\CC}/\mfm_x \in \mod{R/\mfm_p}$, so we take
\[
\rank_p \Omega^1_{X/\CC} \da \dim_{R/\mfm_p} (\Omega^1_{R/\CC}/k)
\]
where $R/\mfm_p \cong k$ is a fixed field via the map $f\mapsto f(p)$.
:::

:::{.example title="?"}
The previous example is still smooth: we have 
\[
\Omega^1_{R/\CC}/\mfm_p = {\CC dx \oplus \CC dy \over x(p)dx + y(p) dy}
\]
which has $\CC\dash$dimension 1 if we *don't* have $x(p) = y(p) = 0$.
This exactly recovers the Jacobi criterion.
:::

:::{.definition title="$\oxmods$"}
An $\OO_X\dash$module is a sheaf $\mcf$ on $X$ where 

- $\mcf(U) \in \mods{\OO_X(U)}$, so the sections are modules over regular functions, and
- $\mcf(U) \mapsvia{\Res_{UV}} \mcf(V)$ is compatible with the module structure and $\OO_X(U) \mapsvia{\Res_{UV}} \OO_X(V)$, so $\Res_{UV}(f.s) = \Res_{UV}(f) . \Res_{UV}(s)$ for $f\in \OO_X(U)$ and $s\in \mcf(U)$.
:::

:::{.example title="?"}
$\Omega^{1}_{X/\CC}\in \oxmods$, where the sections are 1-forms on open sets, as is $\OO_X$ itself.
An example: $\Omega^1_{\AA^1\smz} = \CC[x,x\inv]dx$.
:::

:::{.example title="?"}
Let $X = \AA^1\slice \CC$, then let $\OO_p$ be the skyscraper sheaf at $p$.
This can be made into an $\OO_X\dash$module in the following way: for $f\in \OO_X(U), s\in \OO_p$, define $f.s = f(p) s$.
How to visualize: think of $\OO_X$ as a trivial bundle.

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-01-17_10-09.pdf_tex} };
\end{tikzpicture}

Compare to the skyscraper sheaf:

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-01-17_10-11.pdf_tex} };
\end{tikzpicture}

:::

:::{.definition title="Morphisms in $\oxmods$"}
If $\mcf, \mcg\in \oxmods$ then $\phi: \mcf\to \mcg$ is a morphism iff it is a morphism of sheaves, so a collection $\phi(U): \mcf(U) \to \mcg(U)$, which are compatible with the module actions.
:::

:::{.example title="?"}
There is a morphism $\OO_X \to \OO_p$ of sheaves determined by
\[
\phi(U): \OO_X(U) &\to \OO_p \\
f &\mapsto f(p)
,\]
which is a morphism in $\oxmods$ since $(g\cdot f)(p) = g(p)\cdot f(p)$ is defined by pointwise multiplication.

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-01-17_10-16.pdf_tex} };
\end{tikzpicture}

Recall that the presheaf $\ker \phi$ is a sheaf, and here $\ker \phi(U) = \ts{f\in \OO_X(U) \st f(p) = 0}$ is the *ideal sheaf* of $p$, $I_p$, so we get a SES
\[
I_p \injects \OO_{\AA^1\slice \CC} \surjects \OO_p
.\]
:::

:::{.definition title="Ideal sheaf"}
For $V \subseteq W$ a subvariety, define
\[
I_V(U) = \ts{f\in \OO_W(U) \st \ro{f}{V \intersect U} = 0}
.\]
:::

:::{.example title="?"}
Let $X = \PP^1\slice \CC$, then recall $\OO_X(X) = \CC$ in this case.
Letting $p\neq q\in X$, there is a map
\[
\OO_X &\to \OO_p \oplus \OO_q \\
f &\mapsto f(p) \oplus f(q)
.\]
This is a surjection of sheaves, despite not being surjective on global sections: $\OO_X(X) = \CC$, while $(\OO_p \oplus \OO_q)(X) = \CC \oplus \CC$.
However, this is an open cover $\mcu \covers X$ where this is a surjection on sections: take $U_1 \da X\smts{p}$ and $U_2\da X\smts{q}$.
Here we get a SES
\[
I_{\ts{p, q}} \injects \OO_{\PP^1\slice \CC}\surjects \OO_p \oplus \OO_q
.\]
:::

:::{.example title="?"}
For $X = \AA^1\slice \CC$, there is an isomorphism 
\[
\Omega^1_{X} &\iso \OO_X \in \oxmods \\
f &\mapsto f\dx
.\]
:::

:::{.remark}
For $X$ not affine, what is $\Omega^1_{X}$?
If $\omega = \sum f_i dx_i$ in one chart and $\sum g_i dy_i$ in another via charts $\vec x, \vec y$, how are they related?
One needs a notion of pullbacks.
We define $\Omega^1_X(U)$ to be well-defined 1-forms $\omega_i \in \Omega^1_X(U \intersect U_i)$ which are compatible on overlaps.
:::

:::{.example title="?"}
Let $X = \PP^1$, glued from affines $U_0 = \spec \CC[s]$ and $U_1 = \spec \CC[t]$ by
\[
t_{01}: U_0 &\to U_1 \\
s &\mapsto t=s\inv
.\]

Take $\omega_i \in \Omega^1_X(U_i)$, then

- $\omega_0 = f_0(s)ds \in \CC[s] ds$
- $\omega_1 = f_1(t)dt \in \CC[t] dt$

Then the compatibility condition is that
\[
t_{01}^*(\omega_1) = f_1(s\inv)d(s\inv) = f_0(s) ds
.\]

This becomes
\[
- {f_1(s\inv) \over s^2}ds &= f_0(s) ds \\
\implies f_1(s\inv) &= -s^2 f_0(s)\\
\implies c_0 + c_1 s\inv + c_2 s^{-2} + \cdots + c_k s^{-k} &= d_0 s^2 + d_1 s^3 + \cdots + d_r s^r
,\]
which can only be true if $f \equiv 0$.
This implies that $\Omega^1_{X}$ is a line bundle.
:::

:::{.definition title="Vector bundle"}
A **line bundle** on $X$ is $\mcf\in \oxmods$ where $\exists \mcu \covers X$ where $\ro{\mcf}{U_i} = \OO_{U_i}$.
A **vector bundle** of rank $r$ is such an $\mcf$ where $\ro{\mcf}{U_i} = \OO_{U_i}\sumpower{r}$ for some $r$.
:::

:::{.example title="?"}
$\OO_p$ is not a vector bundle, since $\OO_p(U)\not\iso \OO_X(U)\sumpower{r}$ for any $r$ or any affine open $U\ni p$.
:::

:::{.definition title="Divisors"}
A Weil divisor on $X$ is a $\ZZ\dash$linear combination of irreducible codimension 1 subvarieties.
For $D = \sum n_i p_i$, its degree is $\sum n_i$.
:::

:::{.example title="?"}
The irreducible codimension 1 subvarieties of $\PP^1$ are points, so 
\[
\WDiv(\PP^1) = \bigoplus _{p\in \PP^1} \ZZ[p]
.\]
For example, $\WDiv(\PP^1\slice\CC)\ni D \da 2[0] - [\pi] + 3[\infty]$ and $\deg D = 4$.
Similarly, $\WDiv(\AA^2)\ni [V(x)] - [V(y)]$.
:::

:::{.definition title="Divisors of functions"}
Let $X$ be irreducible and $f\in \OO_X(U)$ with $U \subseteq X$ Zariski open, and define $\div(f) \da \sum n_i [D_i]$ where $n_i$ is the order of zeros/poles of $f$ along $D_i$.
:::

:::{.example title="?"}
Let $x/y^2 \in \OO_{\AA^2}(V(y)^c)$, then $\div(x/y^2) = [V(x)] - 2[V(y)]$.
Similarly, $f\da {s^2-t^2\over st}\in \OO_{\PP^1}(\PP^1\smts{0, \infty})$ has divisor $\div(f) = [1] + [-1] - [0] - [\infty]$.
:::