%theorems and the like


\theoremstyle{plain}

\newtheorem{thm}{Theorem}[section]
\newtheorem{thmm}[thm]{Main Theorem}
\newtheorem{pro}[thm]{Proposition}
\newtheorem{prb}[thm]{Problem}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{alm}[thm]{Ampleness lemma}
\newtheorem{nvlem}[thm]{Nonvanishing lemma}
\newtheorem{cla}[thm]{Claim}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{con}[thm]{Conjecture}
\newtheorem{fgcon}[thm]{Finite generation Conjecture}
\newtheorem{cbs}[thm]{CBS conjecture}
\newtheorem{ass}[thm]{Assumption}
\newtheorem{concl}[thm]{Conclusion}
\newtheorem{step}{Step}
\newtheorem{case}{Case}

\theoremstyle{definition}

\newtheorem{dfn}[thm]{Definition}
\newtheorem{ntn}[thm]{Notation and Conventions}
\newtheorem{nt}[thm]{Notation}
\newtheorem{conv}[thm]{Convention}
\newtheorem{dfncor}[thm]{Definition-Corollary}
\newtheorem{dfnthm}[thm]{Definition-Theorem}
\newtheorem{dfnlm}[thm]{Definition-Lemma}
\newtheorem{rem}[thm]{Remark}
\newtheorem{ter}[thm]{Terminology}
\newtheorem{exa}[thm]{Example}
\newtheorem{setup}[thm]{Set up}
\newtheorem{say}[thm]{}
\newtheorem{cau}[thm]{Caution}

\theoremstyle{remark}

\newtheorem{exe}[thm]{Exercise}
%\newtheorem{prb}[thm]{Problem}
\newtheorem{hop}[thm]{Hope}
\newtheorem{que}[thm]{Question}




%commands
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\PS}{\mathbb{P}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\QQ}{\mathcal{Q}}
\newcommand{\A}{\mathbb{A}}


%operators

\DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Ker}{Ker}
\DeclareMathOperator{\rk}{rk} \DeclareMathOperator{\W}{Weil}
\DeclareMathOperator{\Coker}{Coker}
\newcommand{\dbar}{{\overline{\partial}}}
\DeclareMathOperator{\codim}{Codim}
\DeclareMathOperator{\Exc}{Exc} \DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Gr}{Gr} \DeclareMathOperator{\mult}{mult}
\DeclareMathOperator{\proj}{Proj} \DeclareMathOperator{\Bl}{Bl}
\DeclareMathOperator{\Bs}{Bs}
\DeclareMathOperator{\Def}{Def}
\DeclareMathOperator{\Sing}{Sing}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\HHom}{\mathcal{H}om}
\DeclareMathOperator{\EExt}{\mathcal{E}xt}
\DeclareMathOperator{\defect}{def}
\DeclareMathOperator{\PProj}{\underline{Proj}}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\projL}{proj Lim}
\DeclareMathOperator{\Hilb}{Hilb}
\DeclareMathOperator{\nklt}{nklt}
\DeclareMathOperator{\Ann}{Ann}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\Red}{Red}
