\section{Introduction}

The derived category of coherent sheaves on an algebraic variety is a large and complicated invariant. It contains a lot of information about the variety, and many other invariants may be extracted out of the derived category. Working with this huge invariant directly is difficult, and thus an important notion in this field is the notion of a semiorthogonal decomposition. This is a particular way of decomposing the derived category into smaller pieces. Those pieces are called admissible subcategories.

We know many examples of semiorthogonal decompositions. For instance, a full exceptional collection is nothing but a semiorthogonal decomposition of a category with components equivalent to the derived category of vector spaces. The first example of such a decomposition was given in \cite{beil78} for projective spaces. Full exceptional collections are also known for Grassmannians, del Pezzo surfaces, and other varieties. The set of all semiorthogonal decompositions of a given category has a complicated structure; in particular, it carries the action of the braid group which acts by mutations, introduced in \cite{gorodentsev89,bond-kapr}. There are other tools of various complexity to produce new semiorthogonal decompositions, see, for instance, the survey \cite{kuznetsov-icm}.

Despite a large number of examples, we do not have good structural results for an arbitrary semiorthogonal decomposition. Most of the things we know are ``negative'': for instance, we know that the Jordan--H\"older property does not hold for semiorthogonal decompositions, with counterexamples given in \cite{jordan-holder,kuznetsov-jordanholder}. Another somewhat pathological behavior is the existence of so-called \emph{phantom subcategories}, shown in \cite{gorchinsky-orlov,BGKS-phantoms}, which are admissible subcategories which behave as zero subcategories on the level of $K$-theory.

Among the positive constraints on the structure of admissible subcategories, perhaps the most immediately useful result is proved in \cite{kawatani-okawa}: admissible subcategories are closed under small deformations of objects. The interaction of semiorthogonal decompositions with Hochschild (co)homology has been studied in \cite{kuznetsov-oldhochschild}. Some tools useful for the study of semiorthogonal decompositions are developed in the papers proving the non-existence of nontrivial semiorthogonal decompositions for some classes of varieties \cite{bridgeland,okawa,kawatani-okawa}.

It is generally expected that for sufficiently nice varieties, e.g., for projective spaces, there are more restrictions on the structure of admissible subcategories. For example, it is conjectured in \cite[Rem.~1.7]{kuznetsov-polishchuk} that there are no phantom subcategories in homogeneous spaces. It is surprisingly hard to check these expectations for any variety which is more complicated than the projective line $\P^1$.

In this paper we study admissible subcategories for some surfaces with many anticanonical divisors. Namely, we study the projective plane, rational elliptic surfaces, and del Pezzo surfaces. The strongest theorem is obtained in the case of projective plane. On $\P^2$ we are able to produce a full classification of admissible subcategories. As we prove in Section~\ref{sec: projective plane}, they all turn out to be generated by exceptional collections:

\begin{restatable*}{theorem}{statemaintheoremPlane}
  \label{thm: all admissible subcategories are standard}
  Any admissible subcategory in $\Dbcoh(\P^2)$ is generated by a subcollection of a mutation of the standard exceptional collection $\Dbcoh(\P^2) = \langle \O, \O(1), \O(2) \rangle$.
\end{restatable*}

This theorem gives a classification of admissible subcategories since mutations of the standard exceptional collection in the derived category $\Dbcoh(\P^2)$ have been classified by Gorodentsev and Rudakov in \cite{gorodentsev-rudakov}. In fact, we rely on this classification to prove the theorem above.

The case of $\P^2$ is quite special. In order to study admissible subcategories in other surfaces, we start with admissible subcategories supported set-theoretically on smooth $(-1)$-curves. By the support of an admissible subcategory we just mean the union of set-theoretic supports of all objects in the subcategory. In Section~\ref{sec: minus one curves} we prove a classification for admissible subcategories like that:

\begin{restatable*}{proposition}{statemaintheoremMinusOneCurves}
  \label{prop: new local classification on blow-ups}
  Let $S$ be a smooth proper surface, and let $j\colon E \monoarrow S$ be the embedding of a smooth $(-1)$-curve. Let $\mA \subset \Dbcoh(S)$ be a nonzero admissible subcategory supported on $E$. Then $\mA$ is generated by the exceptional sheaf $j_*\O_E(k)$ for some integer $k \in \Z$.
\end{restatable*}

As an application of this result, we produce some examples of surfaces without phantom subcategories in Corollary~\ref{cor: no phantoms in blow-ups of antifano}, where we show that blow-ups of distinct points on surfaces with globally generated canonical bundles do not contain any phantom subcategories.

In Section~\ref{sec: del pezzos} we consider admissible subcategories in rational elliptic surfaces. For us, as in the paper \cite{heckman-looijenga}, a rational elliptic surface is a smooth proper surface $S$ whose anticanonical linear system $|{-K_S}|$ defines a regular morphism $S \to \P^1$. Any such surface can be obtained by blowing up some subscheme of length nine in the projective plane. We do not have a full classification statement for admissible subcategories in rational elliptic surfaces. However, our methods are at least sufficient to put strong constraints on the possible phantom subcategories.

\begin{restatable*}{theorem}{statemaintheoremRES}
  \label{thm: rational elliptic surfaces}
  Let $\pi\colon S \to \P^1$ be a rational elliptic surface. If $\mB \subset \Dbcoh(S)$ is a phantom subcategory, then the support $\supp(\mB) \subset S$ is contained in the finite union of:
  \begin{itemize}
    \vspace{-\topsep}
    \item sections of the fibration $\pi\colon S \to \P^1$; and
    \item reducible fibers of the morphism $\pi$.
  \end{itemize}
\end{restatable*}

We proceed with a full classification of admissible subcategories, not necessarily phantom ones, whose support is contained in the union of sections of the anticanonical fibration. Note that any section of the fibration $\pi\colon S \to \P^1$ is a smooth $(-1)$-curve in the surface $S$, so Proposition~\ref{prop: new local classification on blow-ups} is the basic case of the classification. In fact, we show that this is essentially the only possibility, and there are no nontrivial admissible subcategories supported on several non-disjoint sections.

\begin{restatable*}{theorem}{statemaintheoremSections}
  \label{thm: admissible subcategories on sections}
  Let $\pi\colon S \to \P^1$ be a rational elliptic surface. Let $\mB \subset \Dbcoh(S)$ be an admissible subcategory such that $\supp(\mB)$ is contained in the union of sections of the map $\pi$. Then there exist finitely many pairwise disjoint sections $\{ E_i \subset S\}_{i \in I}$ and the corresponding integers~$\{ k_i \in \Z \}_{i \in I}$ such that $\mB$ is equal to the subcategory $\bigoplus_{i \in I} \langle \O_{E_i}(k_i E_i) \rangle \subset \Dbcoh(S)$.
\end{restatable*}

For any del Pezzo surface there exists a way to blow-up several points to obtain a rational elliptic surface.  We use the results above to deduce the following consequence for del Pezzo surfaces.

\begin{restatable*}{theorem}{statemaintheoremDelPezzo}
  \label{thm: no phantoms in del pezzos}
  There are no phantom subcategories in del Pezzo surfaces.
\end{restatable*}

The main technical tool that allows us to prove these results is a relation between admissible subcategories and autoequivalences of derived categories of anticanonical divisors, described in Section~\ref{sec: numerical lemmas}. This relation is due to Addington \cite[Prop.~2.1]{addington}. It generalizes the fact that the restriction of an exceptional object to an anticanonical divisor is a spherical object. Addington's result gives amazingly strong structural constraints for the semiorthogonal decompositions of surfaces, and we describe those constraints explicitly in Proposition~\ref{prop: numerical lemma without numbers}. 

\vspace{\baselineskip}

\textbf{Structure of the paper}.
In Section~\ref{sec: preliminaries} we state and prove miscellaneous lemmas about derived categories of coherent sheaves and admissible subcategories. Section~\ref{sec: numerical lemmas} is the technical core of the paper, containing Addington's result and its implications for the constraints that anticanonical divisors put on admissible subcategories. In Section~\ref{sec: projective plane} we prove the classification of admissible subcategories in the derived category of $\P^2$. Section~\ref{sec: minus one curves} contains a classification of admissible subcategories which are supported on a smooth $(-1)$-curve in a surface, and an application for phantom subcategories in some blow-ups. The classification result is used in Section~\ref{sec: del pezzos}, where we study admissible subcategories in rational elliptic surfaces and prove the non-existence of phantoms in del Pezzo surfaces.

\vspace{\baselineskip}

\textbf{Acknowledgements}.
\input{acknowledgements.tex}
{"config": "arxiv", "file": "2006.07643/chapter1.tex"}