Lecture given by George Moulantzikos, notes taken by Edward Pearce, 14/10/2019, 3pm-4:30pm at The University of Sheffield, based on Daniel Huybrechts' book Lectures on K3 surfaces, Chapter 3.

Linear Systems

Let $D$ be a divisor on a variety $X$. We denote by $|D|$ the set of effective divisors linearly equivalent to $D$.

There is a bijection between this set $|D|$ and the projectivization of the set of sections

\[|D| \leftrightarrow \mathbb{P}(\Gamma(X,\mathcal{O}_{X}(D)))\]

To a section $s\in\Gamma(X,\mathcal{O}_{X}(D))$ we can associate the divisor of zeros $(s)_{0}$, which is effective and linearly equivalent to $D$.

If $s, s'\in\Gamma(X,\mathcal{O}_{X}(D))$ are such that $(s)_{0}=(s')_{0}$, then $s=\lambda s'$.

We define the fixed part of a linear system $P$ to be the biggest divisor contained in every element of $P$.

Based points of a linear system $P$.

Maps to projective space $\mathbb{P}^{n}$

There is a bijective correspondence between:

The set of rational maps $\phi:X \dashrightarrow \mathbb{P}^{n}$ such that $\phi(X)$ is not contained in any hyperplane; and
The set of linear systems on $X$ of dimension $n$ with no fixed part.

Given a map $\phi$, $\phi^{\ast}(|H|)$, where $|H|$ is a linear system of hyperplanes in $\mathbb{P}^{n}$, then we can obtain a linear system $P$ and denoting by $\check{P}$ the dual space, we have a map $\phi:X\to\check{P}$ which sends a point $x$ to the set of hyperplanes $H$ in $P$ that contain $x$.

Examples

Projection from a base point $p$ in $\mathbb{P}^{2}$:

For $|H|$ a linear system of hyperplanes which is complete andof dimension 2, then $\{H\ni p\}\subset|H|$

$\phi:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^{1}$ is defined for all $q\neq p$.

Canonical linear system of curves

If the genus $g\ge1$, then $|K|$ is base point free.
If the genus $g\ge3$, then there is a canonical map $\phi_{K}:\mathbb{P}^{2}\to\mathbb{P}^{g-1}$, which is an embedding if and only if $C$ is not hyperelliptic.

Non-hyperelliptic case

$\phi_{K}$ embeds $C$ as a degree $g-1$ projective curve in $\mathbb{P}^{g-1}$.

Hyperelliptic case

$\phi_{K}$ is a composition of a 2:1 map and a Veroneses map, so $\phi_{K}(C)$ is a rational curve of degree $g-1$.

Example

Dimension of quartics in $\mathbb{P}^{2}$ is $\choose{4+2}{2}=14$, $g=3$, $\dim\mathrm{PGL}_{3}=9-1=8$.

A non-hyperelliptic curve is a quartic in $\mathbb{P}^{2}$.

\[K_{C}=(K_{\mathbb{P}^{2}}+C)\big|_{C}=-3H+4H\big|_{C}=H\big|_{C}\]

Riemann-Roch for surfaces - genus formula

Euler characteristic formula - Riemann-Roch formula

\[\chi(\mathcal{O}_{X}(D))=\chi(\mathcal{O}_{X}) + \frac{1}{2}D(D-K_{\ast})\]

As an exercise, verify that:

$\chi(D)=\frac{1}{2}D^{2}$ for abelian surfaces
$\chi(D)=\frac{1}{2}D^{2} + 2$ for K3 surfaces

Curves with negative self-intersection

The case of (-1) curves

Let $S$ be a surface and let $p\in S$. We say that $S'$ is a blow up of $S$ at $p$ if there is a map $\epsilon:S'\to S$ such that the restricton of $\epsilon$$ at

$\epsilon^{-1}(S\setminus\{p\})$ is an isomorphism onto $S\setminus\{p\}$, and $\epsilon^{-1}(p)=E\cong\mathbb{P}^{1}$, which is called the exceptional curve at the blow up.

Castelnuovo's contractibility theorem at the

...

The case of (-2) curves

Let $C$ be a curve in a K3 surface $X$. Then $2g-2=C^{2}$.

Exercise: Prove the genus formula using the adjunction formula

$C$ can be singular, reducible, non-reduced.

The arithmetic genus of a curve is generally defined as $p_{a}(C)=1-\chi(C,\mathcal{O}_{C})$.

Then in this case $2p_{a}(C)-2=C^{2}$.

Exercise: Prove the genus formula using the definition $p_{a}(C)=1-\chi(C,\mathcal{O}_{C})$ and the short exact sequence for Riemann-Roch surfaces given by

\[0\to\mathcal{O}(-C)\to\mathcal{O}_{X}\to\mathcal{O}_{C}\to0\]

Curves on K3 surfaces

Proposition

Let $X$ be a K3 surface and let $C$ be a smooth curve on $X$ of genus $g$. Then:

$C^{2}=2g-2$ and $h^{0}(C)=g+1$.
If $g\ge1$, then $|C|$ is base point free. It defines a morphism $\phi_{C}:X\to\mathbb{P}^{g}$ and the restriction of $\phi_{C}$ on $C$ is the canonical morphism $\phi_{K}:C\to\mathbb{P}^{g-1}$ given by $|K_{C}|$.
If $g=2$, then $\phi:S\to\mathbb{P}^{2}$ is a degree 2 morphism and the branch is a sextic in $\mathbb{P}^{2}$.
If $g=3$, then either
- $\phi$ is a birational morphism and the generic curve of $|C|$ is non-hyperelliptic; or
- $\phi$ is a 2:1 morphism to a rational surface, and the generic curve of $|C|$ is hyperelliptic.
If $g\ge3$ (respectively $g\ge2$), then the morphism defined by $|2C|$ (respectively $|3C|$) is birational.

Examples

Suppose $g=3$, and $\phi(S)\subset\mathbb{P}^{3}$ is a quartic in $\mathbb{P}^{3}$. Then the dimension is 34, $\dim\mathrm{PGL}_{4}=15$, and the difference is 19.

Part 2 by Evgeny Schinder

Very ample, ample, big, and nef line bundles

Let $S$ be a surface and $L=\mathcal{O}(D)$ a line bundle on $S$.

Then very ample implies ample implies nef and big implies nef, where 'nef' stands for numerically effective.

Very ample: $\phi_{D}$ is a closed embedding
Ample: $nD$ is very ample for some $n>>0$
Ample (equivalent condition): $D\cdot C > 0$ for all curves $C\subset X$, and $D^{2}>0$
nef and big: $D\cdot C \ge 0$ for all curves $C\subset X$, and $D^{2}>0$
nef: $D\cdot C \ge 0$ for all curves $C\subset X$.

Examples

$\mathbb{P}^{2}$, then $\mathrm{Pic}\mathbb{P}^{2}=\mathbb{Z}\cdot H$, $D=k\cdot H$.

Then $D$ is very ample iff ample iff nef and big iff $k>0$.

Now consider the case $S=\mathrm{Bl}_{p}\mathbb{P}^{2}$, then $\mathrm{Pic}S=\mathbb{Z}\cdot H \oplus \mathbb{Z}\cdot E$ is generated by the line from $\mathbb{P}^{2}$ and the exceptional curve.

Definition

The positive cone of a K3 surface $X$ is $\mathcal{C}_{X}\subset\{D^{2}>0\}\subset\mathrm{Pic}X\otimes\mathbb{R}=\mathrm{NS}(X)\otimes\mathbb{R}$, where we specifically restrict our attention to the component contatining ample classes.

Proposition

Let $L$ on a K3 $X$, then

$L$ is ample if and only if $L\in\mathcal{C}_{X}$ and for all (-2) curves $C$ we have $L\cdot C>0$.
If $L^{2}\ge0$ and $L\cdot C \ge 0$ for all (-2) curves $C$, then $L$ is nef, unless there are no (-2) curves, in which case either $L$ or $L^{\ast}$ is nef.

Kodaira vanishing theorem

Two big theorems, which both apply to the case $\mathrm{char}(k)=0$.

Theorem (Kodaira vanishing)

Let $X$ be a smooth projective variety and $L$ an ample line bundle. Then

\[\mathrm{H}^{i}(X, L\otimes\omega_{X})=0, \,\forall i>0\]

Example

Let $X=\mathbb{P}^{2}$, then $\omega_{X}=\mathcal{O}(-3)$, and so $L=\mathcal{O}(j)$ is ample if and only if $j\ge1$.

Thus $L\otimes\omega_{X}=\mathcal{O}(j-3)$ where $j-3\ge-2$.

Therefore:

$\mathrm{H}^{1}(X, \mathcal{O}(\ge-2))=0$
$\mathrm{H}^{2}(X, \mathcal{O}(\ge-2))=0$

Similar results hold for $\mathbb{P}^{n}$.

Note that for K3 surfaces, these results hold for any characteristic, and for $L$ big and nef.

Bertini's theorem

Let $X$ be smooth, and $P\subset|L|$ be a linear system. Then a general member of $P$ is smooth outside the base locus of $P$.

Corollary

For $X\subset\mathbb{P}^{N}$, the general hyperplane section is smooth.

Theorem

Let $L$ be a line bundle on a K3 surface $X$ such that

$L^{2}>0$
$|L|\ni C$ contains an irreducible curve

Then $L$ is base point free, and in particular the general member of $|L|$ is smooth.

Irreducible curves on K3 surfaces

Let $C$ be an irreducible curve contained in a K3 surface $X$.

Flow chart: Consider three cases for $C^{2}$:

Negative, then $C^{2}=-2$, and so $C\cong\mathbb{P}^{1}$
Zero, then $C$ is an elliptic curve, i.e. a fiber of $X\to\mathbb{P}^{1}$
Positive - Does there exist a (-2) curve $E$ such that $E\cdot C-0$?
- Yes, then $C$ is big and nef, but not ample
- No, then $C$ is ample

Examples

An example of a line bundle which is nef, but not big and nef

We want $L\cdot C \ge 0$ and $L^{2}=0$.

So take $L=\mathcal{O}(D)$ where $D$ is an eliptic curve.

An example of a line bundle which is big and nef, but not ample

Let $\bar{S}\subset\mathbb{P}^{3}$ be a general quartic with an ordinary double point, and $S\to\bar{S}$ a resolution of singularities.

Then $K_{S}=0$, $\mathrm{Pic}S=\mathbb{Z}\cdot H\oplus \mathbb{Z}\cdot E$, where $H^{2}=4$, $E^{2}=-2$, and $H\cdot E=0$.

Then $H$ is big and nef, but not ample since $H\cdot E=0$. We say that "$\phi_{H}$ contracts $E$"

An example of a line bundle whichis ample, but not very ample

Consider the 2:1 map $\phi_{H}:S\to\mathbb{P}^{2}$ which is branched on a sextic curve. Then $H$ is not very ample, but $H$ is ample.

K3 Surfaces - more on Hodge structures and basic geometry of K3 surfaces

Linear Systems

Maps to projective space \(\mathbb{P}^{n}\)

Examples

Projection from a base point \(p\) in \(\mathbb{P}^{2}\):

Canonical linear system of curves

Non-hyperelliptic case

Hyperelliptic case

Example

Riemann-Roch for surfaces - genus formula

Curves with negative self-intersection

The case of (-1) curves

Castelnuovo's contractibility theorem at the

The case of (-2) curves

Curves on K3 surfaces

Proposition

Examples

Part 2 by Evgeny Schinder

Very ample, ample, big, and nef line bundles

Examples

Definition

Proposition

Kodaira vanishing theorem

Theorem (Kodaira vanishing)

Example

Bertini's theorem

Corollary

Theorem

Irreducible curves on K3 surfaces

Examples

An example of a line bundle which is nef, but not big and nef

An example of a line bundle which is big and nef, but not ample

An example of a line bundle whichis ample, but not very ample