\contentsline {paragraph}{Status of these files}{ii}
\contentsline {section}{\numberline {0.1}A reminder of elementary notions}{1}
\contentsline {section}{\numberline {0.2}Rational transformations}{2}
\contentsline {section}{\numberline {0.3}Surfaces in hyperspaces $S_r$ and resolution of singularities}{5}
\contentsline {section}{\numberline {0.4}Surfaces with normal singularities}{6}
\contentsline {section}{\numberline {0.5}Note}{8}
\contentsline {paragraph}{(1)}{10}
\contentsline {paragraph}{(2)}{11}
\contentsline {subparagraph}{a)}{11}
\contentsline {subparagraph}{b)}{12}
\contentsline {chapter}{\numberline {1}Linear systems of curves}{13}
\contentsline {section}{\numberline {1.1}Linear pencils}{13}
\contentsline {section}{\numberline {1.2}Irrational pencils}{15}
\contentsline {section}{\numberline {1.3}Linear systems}{16}
\contentsline {section}{\numberline {1.4}The characteristic property of linear systems}{18}
\contentsline {section}{\numberline {1.5}Extension of the theorems of Bertini}{19}
\contentsline {section}{\numberline {1.6}The image surfaces of linear systems}{20}
\contentsline {paragraph}{Note}{22}
\contentsline {section}{\numberline {1.7}Equivalent curves and complete linear systems}{22}
\contentsline {section}{\numberline {1.8}Sum and difference of linear systems: the residue theorem}{25}
\contentsline {paragraph}{Note}{26}
\contentsline {paragraph}{Remark}{27}
\contentsline {paragraph}{NOTE}{27}
\contentsline {section}{\numberline {1.9}The image surface of the sum system}{28}
\contentsline {section}{\numberline {1.10}Virtual characters}{29}
\contentsline {section}{\numberline {1.11}Exceptional curves}{31}
\contentsline {section}{\numberline {1.12}A note on the reducible exceptional curves}{35}
\contentsline {section}{\numberline {1.13}The virtual characteristic series}{38}
\contentsline {paragraph}{Note}{38}
\contentsline {chapter}{\numberline {2}Covariant and invariant systems}{39}
\contentsline {section}{\numberline {2.1}Jacobian curves}{39}
\contentsline {section}{\numberline {2.2}The Jacobian system}{40}
\contentsline {paragraph}{Note}{41}
\contentsline {section}{\numberline {2.3}The fundamental theorem}{45}
\contentsline {section}{\numberline {2.4}Canonical curves}{47}
\contentsline {paragraph}{Note}{49}
\contentsline {section}{\numberline {2.5}Properties of canonical curves}{49}
\contentsline {paragraph}{Note}{50}
\contentsline {section}{\numberline {2.6}Geometric genus and linear genus}{50}
\contentsline {paragraph}{Note}{53}
\contentsline {section}{\numberline {2.7}The exceptional curves as fixed parts of the canonical system}{54}
\contentsline {paragraph}{NOTE I}{55}
\contentsline {paragraph}{NOTE II}{56}
\contentsline {paragraph}{EXERCISE}{57}
\contentsline {section}{\numberline {2.8}Bicanonical and pluricanonical curves}{57}
\contentsline {section}{\numberline {2.9}A historical note}{59}
\contentsline {chapter}{\numberline {3}Adjoint surfaces}{63}
\contentsline {section}{\numberline {3.1}Surfaces of order $n-3$ adjoint to a surface of order $n$ without proper multiple points}{63}
\contentsline {section}{\numberline {3.2}Adjoint surfaces of arbitrary order}{65}
\contentsline {section}{\numberline {3.3}Ruled surfaces}{66}
\contentsline {section}{\numberline {3.4}Characteristic properties of canonical curves and of the curves adjoint to a linear system}{67}
\contentsline {section}{\numberline {3.5}Isolated multiple points}{68}
\contentsline {section}{\numberline {3.6}Adjoints and subadjoints}{69}
\contentsline {paragraph}{EXAMPLES}{70}
\contentsline {section}{\numberline {3.7}Multiple points of the first kind}{71}
\contentsline {paragraph}{EXAMPLES}{72}
\contentsline {paragraph}{EXERCISE}{72}
\contentsline {section}{\numberline {3.8}Double planes}{73}
\contentsline {section}{\numberline {3.9}The genus of the fundamental curves and their influence on the adjoints}{75}
\contentsline {section}{\numberline {3.10}Point singularities of a higher kind}{77}
\contentsline {paragraph}{Remark}{80}
\contentsline {paragraph}{EXERCISE}{83}
\contentsline {paragraph}{Remark}{84}
\contentsline {section}{\numberline {3.11}Bicanonical curves and biadjoint surfaces}{84}
\contentsline {section}{\numberline {3.12}Surfaces of genus $p=0$ and bigenus $P>0$}{87}
\contentsline {paragraph}{Remark}{90}
\contentsline {section}{\numberline {3.13}Equivalence criteria}{91}
\contentsline {paragraph}{Remark}{94}
\contentsline {chapter}{\numberline {4}The numerical genus and the Riemann--Roch theorem for surfaces}{96}
\contentsline {section}{\numberline {4.1}Introduction}{96}
\contentsline {section}{\numberline {4.2}Conditions imposed on a surface that contains a curve: postulation formulas}{97}
\contentsline {section}{\numberline {4.3}Dimension of the systems of adjoint surfaces to a given one}{102}
\contentsline {section}{\numberline {4.4}The numerical genus}{104}
\contentsline {section}{\numberline {4.5}Complements}{108}
\contentsline {paragraph}{Lemma on the completeness of the sum of two series belonging to a curve.}{111}
\contentsline {section}{\numberline {4.6}Remarks on regularity conditions for a surface}{112}
\contentsline {paragraph}{Note}{115}
\contentsline {section}{\numberline {4.7}Examples of irregular surfaces}{115}
\contentsline {paragraph}{Remark}{118}
\contentsline {section}{\numberline {4.8}The theorem of Riemann--Roch for surfaces: systems larger than the canonical}{118}
\contentsline {section}{\numberline {4.9}Elimination of the exceptional curves}{121}
\contentsline {section}{\numberline {4.10}Completeness of the series cut out by a small linear system on the curves of a large system}{123}
\contentsline {section}{\numberline {4.11}The theorem of Riemann--Roch for arbitrary linear systems}{126}
\contentsline {section}{\numberline {4.12}The deficiency of the characteristic series}{128}
\contentsline {paragraph}{Remark}{131}
\contentsline {section}{\numberline {4.13}Historical note}{132}
\contentsline {section}{\numberline {4.14}Virtual curves}{136}
\contentsline {section}{\numberline {4.15}Regular and superabundant systems}{141}
\contentsline {section}{\numberline {4.16}More on the regularity of the adjoint system of a curve on a regular surface}{144}
\contentsline {section}{\numberline {4.17}The adjoint system of a connected curve consisting of multiple components on a regular surface (continued)}{146}
\contentsline {section}{\numberline {4.18}On the reducibility of the canonical system}{150}
\contentsline {paragraph}{Note}{152}
\contentsline {section}{\numberline {4.19}On the regularity of the bicanonical system}{153}
\contentsline {paragraph}{(1)}{153}
\contentsline {paragraph}{(2)}{155}
\contentsline {paragraph}{(3)}{157}
\contentsline {section}{\numberline {4.20}Remark on the irreducibility of the bicanonical system}{157}
\contentsline {section}{\numberline {4.21}Genera of a reducible surface}{158}
\contentsline {chapter}{\numberline {5}Numerical invariants and multiple planes}{161}
\contentsline {section}{\numberline {5.1}The Zeuthen--Segre invariant}{161}
\contentsline {paragraph}{Remark}{163}
\contentsline {section}{\numberline {5.2}Irrational pencil}{166}
\contentsline {section}{\numberline {5.3}Expressing the Zeuthen--Segre invariant in terms of the genera}{167}
\contentsline {paragraph}{Note}{170}
\contentsline {paragraph}{Remark}{171}
\contentsline {section}{\numberline {5.4}Cuspidal curves of a net}{172}
\contentsline {section}{\numberline {5.5}Historical note}{174}
\contentsline {section}{\numberline {5.6}Characters of a net and multiple planes}{175}
\contentsline {paragraph}{Remark}{177}
\contentsline {section}{\numberline {5.7}$(1,n)$ correspondence between two surfaces: double planes}{177}
\contentsline {section}{\numberline {5.8}Correspondence formulas}{179}
\contentsline {paragraph}{EXAMPLES}{183}
\contentsline {paragraph}{Remark}{184}
\contentsline {section}{\numberline {5.9}The existence theorem}{186}
\contentsline {section}{\numberline {5.10}Complements: limit forms of the branch curve according to Chisini}{192}
\contentsline {section}{\numberline {5.11}The moduli of a class of algebraic surfaces}{197}
\contentsline {section}{\numberline {5.12}Digression on the completeness of\\ the characteristic series of a complete\\ system of plane curves with nodes and\\ cusps}{200}
\contentsline {section}{\numberline {5.13}The moduli of the regular surfaces of genus $p>3$}{202}
\contentsline {section}{\numberline {5.14}Historical note and complements}{206}
\contentsline {chapter}{\numberline {6}Regular surfaces: the minimum for the genera and rationality conditions}{209}
\contentsline {section}{\numberline {6.1}The lower bound for the linear genus}{209}
\contentsline {section}{\numberline {6.2}The bigenus and the plurigenera}{212}
\contentsline {paragraph}{Remark}{215}
\contentsline {section}{\numberline {6.3}Examples: double planes with linear genus $p^{(1)}=1$}{216}
\contentsline {section}{\numberline {6.4}Rationality conditions for a surface}{222}
\contentsline {paragraph}{Hypothesis I.}{223}
\contentsline {paragraph}{Hypothesis II}{226}
\contentsline {chapter}{\numberline {7}Classification of surfaces of linear genus $p^{(1)}=1$}{228}
\contentsline {section}{\numberline {7.1}Surfaces of genus zero and bigenus one, with bicanonical curve of order zero}{228}
\contentsline {paragraph}{Remark}{237}
\contentsline {section}{\numberline {7.2}Surfaces with all the genera equal to one}{238}
\contentsline {paragraph}{(1)}{243}
\contentsline {paragraph}{(2)}{244}
\contentsline {paragraph}{(3)}{244}
\contentsline {paragraph}{Remark}{245}
\contentsline {section}{\numberline {7.3}A historical note}{245}
\contentsline {section}{\numberline {7.4}Surfaces of linear genus $p^{(1)}=1$ with $P>1$}{248}
\contentsline {paragraph}{Note}{256}
\contentsline {chapter}{\numberline {8}Regular canonical and pluricanonical surfaces}{258}
\contentsline {section}{\numberline {8.1}Introduction}{258}
\contentsline {section}{\numberline {8.2}Canonical surfaces of genus $p=4$ and linear genus $p^{(1)}=6,5$}{259}
\contentsline {section}{\numberline {8.3}Surfaces with $p=4$ and $p^{(1)}=7$}{262}
\contentsline {paragraph}{Note}{262}
\contentsline {section}{\numberline {8.4}Surfaces with $p=4$ and $p^{(1)}=8$}{264}
\contentsline {paragraph}{Note}{270}
\contentsline {section}{\numberline {8.5}Surfaces with $p=4$ and $p^{(1)}=9$}{271}
\contentsline {paragraph}{Note}{273}
\contentsline {section}{\numberline {8.6}Surfaces with $p=4$ and $p^{(1)}>9$}{274}
\contentsline {section}{\numberline {8.7}Canonical surfaces with $p=5$ and $p^{(1)}=9$}{275}
\contentsline {section}{\numberline {8.8}Surfaces with $p=5$ and $p^{(1)}=10$}{276}
\contentsline {paragraph}{QUESTIONS}{279}
\contentsline {paragraph}{Note}{279}
\contentsline {section}{\numberline {8.9}Surfaces on a variety of dimension 3 or more}{279}
\contentsline {section}{\numberline {8.10}Examples of canonical surfaces in hyperspace}{282}
\contentsline {section}{\numberline {8.11}Minimal value of the linear genus with respect to the superficial genus}{285}
\contentsline {section}{\numberline {8.12}Upper bound for the linear genus $p^{(1)}$ in terms of the superficial genus}{289}
\contentsline {section}{\numberline {8.13}Canonical systems belonging to an involution}{290}
\contentsline {section}{\numberline {8.14}Surfaces with linear genus $p^{(1)}=2$: first case $p=2$}{294}
\contentsline {paragraph}{Remark}{295}
\contentsline {section}{\numberline {8.15}Surfaces with $p^{(1)}=2$ and $p=1$ ($P=3$, $P_3=5$)}{295}
\contentsline {paragraph}{Remark}{297}
\contentsline {section}{\numberline {8.16}Surfaces with $p^{(1)}=2$ and $p=0$ ($P=2$, $P_3=4$)}{298}
\contentsline {paragraph}{Remark}{299}
\contentsline {section}{\numberline {8.17}Surfaces with linear genus $p^{(1)}=3$: first case $p=3$}{302}
\contentsline {section}{\numberline {8.18}Surfaces with $p^{(1)}=3$ and $p=2$ ($P=5$)}{303}
\contentsline {paragraph}{Remark}{306}
\contentsline {section}{\numberline {8.19}Surfaces with $p^{(1)}=3$ and $p=1$ ($P=4$)}{307}
\contentsline {section}{\numberline {8.20}Surfaces with $p=0$ and $p^{(1)}=3$}{311}
\contentsline {section}{\numberline {8.21}Simple and multiple pluricanonical surfaces}{312}
\contentsline {chapter}{\numberline {9}Irregular surfaces and continuous systems of\\ inequivalent curves}{315}
\contentsline {section}{\numberline {9.1}Introduction}{315}
\contentsline {section}{\numberline {9.2}The arithmetic condition for a curve on a surface with geometric irregularity $q$ to belong to an $\infty ^q$ continuous series of inequivalent curves}{317}
\contentsline {section}{\numberline {9.3}The Picard variety corresponding to an irregular surface}{318}
\contentsline {section}{\numberline {9.4}The characteristic property of irregular algebraic surfaces: the fundamental theorem for $p_g=0$}{319}
\contentsline {section}{\numberline {9.5}The fundamental theorem for $p_g>0$}{320}
\contentsline {section}{\numberline {9.6}The history of the theory of continuous systems}{329}
\contentsline {section}{\numberline {9.7}On various attempts to prove and extend the fundamental theorem}{336}
\contentsline {paragraph}{Remark}{342}
\contentsline {section}{\numberline {9.8}The paracanonical system}{343}
\contentsline {paragraph}{Remark}{346}
\contentsline {section}{\numberline {9.9}A digression on Abelian varieties}{346}
\contentsline {paragraph}{Note}{350}
\contentsline {section}{\numberline {9.10}Continuation: the genus of the Abelian varieties}{350}
\contentsline {paragraph}{Note}{352}
\contentsline {section}{\numberline {9.11}Irrational pencil on the surfaces with geometric genus zero}{353}
\contentsline {paragraph}{Note}{357}
\contentsline {section}{\numberline {9.12}A note on the surfaces of irregularity one}{357}
\contentsline {chapter}{\numberline {10}Surfaces with geometric genus zero}{359}
\contentsline {section}{\numberline {10.1}Introduction}{359}
\contentsline {section}{\numberline {10.2}Surfaces containing a system of curves of genus $\pi $ and degree $n>2\pi -2$}{360}
\contentsline {paragraph}{Note}{362}
\contentsline {paragraph}{Note}{364}
\contentsline {section}{\numberline {10.3}A lemma on the reducible curves in a pencil}{364}
\contentsline {section}{\numberline {10.4}Surfaces with genus $p_g=0$ and $p_a<-1$}{366}
\contentsline {section}{\numberline {10.5}Surfaces $F$ with $p_g=0$ and $p_a=-1$, possessing an elliptic pencil of curves $K$ of genus $\pi >1$: Lemma I}{367}
\contentsline {section}{\numberline {10.6}Lemma II: a sketch of the proof}{369}
\contentsline {section}{\numberline {10.7}The first case: the genus $\pi $ of the curves $K$ is even}{372}
\contentsline {paragraph}{Remark}{377}
\contentsline {section}{\numberline {10.8}The second case: the genus $\pi $ of the curves $K$ is odd}{378}
\contentsline {section}{\numberline {10.9}Lemma III}{380}
\contentsline {section}{\numberline {10.10}Conclusion: the surfaces $F$ also possess a linear pencil of elliptic curves}{381}
\contentsline {section}{\numberline {10.11}Surfaces with genera $p_g=0$ and $p_a=-1$ with an elliptic pencil of curves of genus $\pi =1$}{382}
\contentsline {paragraph}{Note}{384}
\contentsline {section}{\numberline {10.12}Elliptic surfaces}{386}
\contentsline {paragraph}{Remark}{388}
\contentsline {section}{\numberline {10.13}Construction of the elliptic surfaces of genus $p_g=0$}{389}
\contentsline {section}{\numberline {10.14}Surfaces with all plurigenera equal to zero: characterization of the ruled surfaces}{393}
\contentsline {paragraph}{Remark}{398}
\contentsline {section}{\numberline {10.15}A historical note}{400}
\contentsline {section}{\numberline {10.16}Surfaces with pluricanonical curves of order zero: the types with elliptic curves $K$ with general modulus}{402}
\contentsline {section}{\numberline {10.17}Continuation: The harmonic case}{405}
\contentsline {section}{\numberline {10.18}The equianharmonic case}{409}
\contentsline {section}{\numberline {10.19}Summary}{411}
\contentsline {paragraph}{Note}{412}
\contentsline {chapter}{\numberline {11}The general classification of surfaces}{414}
\contentsline {section}{\numberline {11.1}Introduction}{414}
\contentsline {section}{\numberline {11.2}Surfaces admitting a continuous series of birational transformations into themselves: the cases leading to ruled surfaces}{416}
\contentsline {section}{\numberline {11.3}Elliptic and hyperelliptic surfaces}{420}
\contentsline {paragraph}{Case\nobreakspace {}1}{421}
\contentsline {paragraph}{Note}{421}
\contentsline {paragraph}{Case\nobreakspace {}2}{422}
\contentsline {section}{\numberline {11.4}Characterization of elliptic and hyperelliptic surfaces by means of the values of the genera}{425}
\contentsline {paragraph}{Remark}{431}
\contentsline {paragraph}{Note}{432}
\contentsline {section}{\numberline {11.5}Surfaces with genera $p_g=1$ and $p^{(1)}=1$}{433}
\contentsline {paragraph}{Remark}{437}
\contentsline {section}{\numberline {11.6}A note on the geometric theory of hyperelliptic surfaces}{438}
\contentsline {section}{\numberline {11.7}The general classification of algebraic surfaces}{442}
\contentsline {paragraph}{(A) $P_{12}=0$}{443}
\contentsline {paragraph}{$(\@mathrm A')$}{443}
\contentsline {paragraph}{$(\@mathrm A'')$}{443}
\contentsline {paragraph}{(B) $P_{12}=1$, $p^{(1)}=1$ ($p_a\ge -1$)}{443}
\contentsline {paragraph}{$(\@mathrm B')$}{443}
\contentsline {paragraph}{$(\@mathrm B'')$}{444}
\contentsline {paragraph}{(C) $P_{12}>1$ and $p^{(1)}=1$}{444}
\contentsline {paragraph}{(D) $P_{12}$, $p^{(1)}>1$}{445}
\contentsline {section}{\numberline {11.8}Surfaces with negative arithmetic genus}{446}
\contentsline {paragraph}{Note}{448}
\contentsline {section}{\numberline {11.9}Summary of the above classification}{450}
\contentsline {chapter}{Index}{453}