Complutense University of Madrid doctoral thesis, Mathematical Research (svmonoucm)

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This work has two well different parts. The first one deals with Milnor number, Tjurina number, spectral values and the geometric genus which constitute the main topological and analytic of an isolated hypersurface singularity. One of the main contributions of the first part of this memoir has been to motivate the study of the relations between the Milnor number, Tjurina number, spectral values and the geometric genus from a new point of view. This certainly provides a new way of understanding the importance of Durfee conjecture in the context of isolated hypersurface singularities. In conclusion, the first part of the memoir presents a new research program in order to gain understanding of the interplay between the main topological and analytic invariants of a hypersurface singularity.


The second part deals with numerical semigroups and its combinatorics. The main results in this part can be split into two parts: one devoted to Wilf's conjecture on numerical semigroups and its generalization to modules over them; and the other one about value set of modules over the local ring of an irreducible plane curve singularities with one Puiseux pair. Our main contribution to Wilf's conjecture has been to propose and initiate its extension to modules over a numerical semigroup. As a consequence, we have provided new insight in some related problems to Wilf's conjecture. Our contributions in the case of value set of modules over the local ring of an irreducible plane curve singularities with one Puiseux pair has been to provide the first step into the generalization to modules over a numerical semigroup of a Theorem of Bresinsky and Teissier about the value semigroup of an irreducible plane curve singularity. As a consequence, we will deduce some new features about the value set of K\"{a}hler differentials of an irreducible plane curve singularity with one Puiseux pair.

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% sample preface
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%% Please write your preface here
The abstract

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% sample preface
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Resumen en español

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\extrachap{Acknowledgments}
\addcontentsline{toc}{chapter}{Acknowledgments}
\setcounter{page}{1}



Agradecimientos. 


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En esta página puede incluirse un epígrafe si se quiere incluir una cita literaria. No es obligatorio. 

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\chapter{Un apendice}
Un apendice

%    Insert the bibliography data

\bibliographystyle{}
\begin{thebibliography}{99.}
\bibitem{zariski} P. Samuel, O. Zariski, \textit{Commutative algebra, Volume I, With the cooperation of I. S. Cohen}, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey (1958).





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%    amsalpha, or (for "historical" overviews) natbib style.

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\begin{dedication}
Use the template \emph{dedic.tex} together with the Springer document class SVMono for monograph-type books or SVMult for contributed volumes to style a quotation or a dedication\index{dedication} at the very beginning of your book in the Springer layout
\end{dedication}

% framed.sty   v 0.96    2011/10/22
% Copyright (C) 1992-2011 by Donald Arseneau  ([email protected])
% These macros may be freely transmitted, reproduced, or modified
% for any purpose provided that this notice is left intact.
% 
%====================== Begin Instructions =======================
%
%                       framed.sty
%                       ~~~~~~~~~~
% Create framed, shaded, or differently highlighted regions that can 
% break across pages.  The environments defined are 
%   framed     - ordinary frame box (\fbox) with edge at margin
%   oframed    - framed with open top/bottom at page breaks
%   shaded     - shaded background (\colorbox) bleeding into margin
%   shaded*    - shaded background (\colorbox) with edge at margin
%   snugshade  - shaded with tight fit around text (esp. in lists)
%   snugshade* - like snugshade with shading edge at margin
%   leftbar    - thick vertical line in left margin
% 
% to be used like
% \begin{framed}
%  copious text
% \end{framed}
%
% But the more general purpose of this package is to facilitate the
% definition of new environments that take multi-line material,
% wrap it with some non-breakable formatting (some kind of box or
% decoration) and allow page breaks in the material.  Such environments
% are defined to declare (or use) \FrameCommand for applying the boxy 
% decoration, and \MakeFramed{settings} ... \endMakeFramed wrapped
% around the main text argument (environment body).
%
% The "framed" environment uses "\fbox", by default, as its "\FrameCommand"
% with the additional settings "\fboxrule=\FrameRule" and "\fboxsep=\FrameSep".
% You can change these lengths (using "\setlength") and you can change 
% the definition of "\FrameCommand" to use much fancier boxes.  
%
% In fact, the "shaded" environment just redefines \FrameCommand to be
% "\colorbox{shadecolor}" (and you have to define the color `"shadecolor"':
% "\definecolor{shadecolor}...").
%
% Although the intention is for other packages to define the varieties
% of decoration, a command "\OpenFbox" is defined for frames with open 
% tops or bottoms, and used for the "oframed" environment.  This facility
% is based on a more complex and capable command "\CustomFBox" which can
% be used for a wider range of frame styles.  One such style of a title-bar
% frame with continuation marks is provided as an example.  It is used by
% the "titled-frame" environment.  To make use of "titled-frame" in your
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% definitions, you must define the colors TFFrameColor (for the frame)
% and a contrasting TFTitleColor (for the title text).
%
% A page break is allowed, and even encouraged, before the framed
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% be separated from the body.
%
% The contents of the framed regions are restricted: 
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% This package will not work with the page breaking of multicol.sty,
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%
% The MakeFramed environment does the work.  Its `settings' argument
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% or something similar; for instance, the snugshade environment uses
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% "\hspace" in "\FrameCommand" to reproduce the list margin ouside the
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%
% There are actually four variants of "\FrameCommand" to allow different
% formatting for each part of an environment broken over pages.  Unbroken
% text is adorned by "\FrameCommand", whereas split text first uses 
% "\FirstFrameCommand", possibly followed by "\MidFrameCommand", and
% finishing with "\LastFrameCommand".  The default definitions for 
% these three just invokes "\FrameCommand", so that all portions are
% framed the same way.  See the oframe environment for use of distinct
% First/Mid/Last frames.
%
% Expert commands:
% \MakeFramed, \endMakeFramed: the "MakeFramed" environment
% \FrameCommand: command to draw the frame around its argument
% \FirstFrameCommand: the frame for the first part of a split environment
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% \FrameHeightAdjust: macro; height of frame above baseline at top of page
% \OuterFrameSep: vertical space before and after the framed env. Defaults to "\topsep"
%
% This is still a `pre-production' version because I can think of many
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% ToDo: 
% Test more varieties of list 
% Improve and correct documentation
% Propagation of \marks
% Handle footnotes (how??) floats (?) and marginpars.
% Stretchability modification.
% Make inner contents height/depth influence placement.
%======================== End Instructions ========================

\ProvidesPackage{framed}[2011/10/22 v 0.96: 
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\newenvironment{framed}% using default \FrameCommand
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%
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  \def\LastFrameCommand{\fboxsep8pt\fboxrule2pt
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% The title bar for \TitleBarFrame
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%%%   {\showoutput\showbox\@tempboxa}%
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   \fb@put@frame\FrameCommand\FirstFrameCommand
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% \fb@put@frame takes the contents of \@tempboxa and puts all, or a piece,
% of it on the page with a frame (\FrameCommand, \FirstFrameCommand,
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|   \message{=============== Entering putframe ====================^^J
|     \pagegoal=\the\pagegoal,  \pagetotal=\the\pagetotal. }%
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 \else
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        \edef\fb@resto@set{\boxmaxdepth\the\boxmaxdepth 
                           \splittopskip\the\splittopskip}%
        \boxmaxdepth\z@ \splittopskip\z@
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|       \toks99\expandafter{\splitfirstmark}%
|       \toks98\expandafter{\splitbotmark}%
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|       \message{Natural height of split box is \the\ht\tw@, leaving 
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        \advance\dimen@\topskip
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           % need work here???
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           \fb@adjheight
           \fb@put@frame#1#2% INSERTED ???
        \else % Got material split off at the head
           \egroup \fb@resto@set
           \ifvoid\@tempboxa % it all fit after all
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              \fb@putboxa#1%
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                \centerline{#2{\box\tw@}}%  ??? \centerline bad idea
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                \box\tw@
              \fi
              \hrule \@height\z@ \@width\hsize
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  \fi\fi\fi\fi\fi
}

\def\fb@putboxa#1{%
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    \PackageWarning{framed}{Boxa is void -- discard it. }%
  \else
|   \message{Frame and place boxa. }%
|   %{\showoutput\showbox\@tempboxa}%
    \centerline{#1{\box\@tempboxa}}%
  \fi
}

\def\fb@afterframe{%
    \nointerlineskip \null %{\showoutput \showlists}
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}

% measure width and height added by frame (#1 = frame command)
% call results \fb@frw and \fb@frh
% todo: a mechanism to handle wide frame titles 
\newdimen\fb@frw
\newdimen\fb@frh
\def\fb@sizeofframe#1{\begingroup
 \setbox\z@\vbox{\vskip-5in \hbox{\hskip-5in 
   #1{\hbox{\vrule \@height 4.7in \@depth.3in \@width 5in}}}%
   \vskip\z@skip}%
|  \message{Measuring frame addition for \string#1 in \@currenvir\space 
|    gives ht \the\ht\z@\space and wd \the\wd\z@. }%
|  %{\showoutput\showbox\z@}%
 \global\fb@frw\wd\z@ \global\fb@frh\ht\z@
 \endgroup
}

\def\fb@adjheight{%
  \vbox to\FrameHeightAdjust{}% get proper baseline skip from above.
  \penalty\@M \nointerlineskip
  \vskip-\FrameHeightAdjust
  \penalty\@M} % useful for tops of pages

\edef\zero@glue{\the\z@skip}

\catcode`\|=\FrameRestore

% Provide configuration commands:
\providecommand\FrameCommand{%
 \setlength\fboxrule{\FrameRule}\setlength\fboxsep{\FrameSep}%
 \fbox}
\@ifundefined{FrameRule}{\newdimen\FrameRule \FrameRule=\fboxrule}{}
\@ifundefined{FrameSep} {\newdimen\FrameSep  \FrameSep =3\fboxsep}{}
\providecommand\FirstFrameCommand{\FrameCommand}
\providecommand\MidFrameCommand{\FrameCommand}
\providecommand\LastFrameCommand{\FrameCommand}

% Height of frame above first baseline when frame starts a page:
\providecommand\FrameHeightAdjust{6pt}

% \FrameRestore has parts of \@parboxrestore, performing a similar but 
% less complete restoration of the default layout.  See how it is used in
% the "settings" argument of \MakeFrame.  Though not a parameter, \hsize 
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% frame before invoking \FrameRestore.  
\def\FrameRestore{%
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          \ifdim\rightmargin>\z@ 1\fi
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     %     \message{In a list: \linewidth=\the\linewidth, \@totalleftmargin=\the\@totalleftmargin,
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     \@setminipage % snug fit around the item.  I would like this to be non-global.
     % Now try to propageate changes of width from \hsize to list parameters.
     % This is deficient, but a more advanced way to indicate modification to text 
     % dimensions is not (yet) provided; in particular, no separate left/right
     % adjustment.
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   \else % Not in list
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   \fi
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}

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\Extrachap{Introduction}


Introducción

\begin{center}

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{\Large \textbf {Universidad Complutense de Madrid }}

\bigskip 

{\Large Facultad de Ciencias Matem\'aticas}

\medskip 

{\Large Programa de Doctorado en Investigación Matemática}

\medskip 

{\Large  Instituto (ICMAT, IMI, ...). Poner sólo si se pertenece a alguno}


\bigskip 

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\includegraphics[width=0.2\textwidth]{{UCM.eps}}

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{\Large \textbf{TESIS DOCTORAL}} 


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 {\Huge 
 
  Title of the thesis. 
  
  Título de la tesis 
  
  }%     % Title set in \Huge size.

 
\bigskip 

\bigskip 
		

{\large Memoria para optar al grado de doctor presentada por }


\bigskip 

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  {\LARGE                Nombre y apellidos completo del doctorando/a  }%
  
  
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{\large Director/a/es: }


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{\Large Nombre y apellidos completos de los directores}		
		
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{\large Año}
	
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\chapter*{List of Publications}
\vspace{-1cm}
The main results of this memoir have given rise to the following research articles:

\begin{enumerate}
	\item[] Chapter \ref{chap:minimaltau}
	\\
	\subitem 1. P.Almirón, G. Blanco: \textbf{A note on a question of Dimca and Greuel}, C.R. Acad. Sci. Paris, Ser. I (2019). \href{https://doi.org/10.1016/j.crma.2019.01.002}{DOI: 10.1016/j.crma.2019.01.002}.
	
	\subitem 2. M.Alberich-Carramiñana, P. Almirón, G.Blanco, A. Melle-Hernández: \textbf{The minimal Tjurina number of irreducible germs of plane curve singularities}. Indiana Univ. Math. J. 70 No. 4 (2021), 1211–1220. \href{https://doi.org/10.1512/iumj.2021.70.8583}{DOI: 10.1512/iumj.2021.70.8583}.
	
	\subitem 3. P. Almirón: \textbf{The 4/3 problem for germs of isolated plane curve singularities}. In Extended Abstracts GEOMVAP 2019. Birkhäuser. Trends in Mathematics. 2021, 15, 139-143.\href{ https://doi.org/10.1007/978-3-030-84800-2_23}{DOI: \(10.1007/978-3-030-84800-2\_23\)}
	\\
	\item [] Chapter \ref{chap:mutau}
	\\
	\subitem 4. P. Almirón: \textbf{On the quotient of Milnor and Tjurina numbers for two-dimensional isolated hypersurface singularities}. To appear in Mathematische Nachrichten. \href{https://arxiv.org/pdf/1910.12843.pdf}{arxiv:1910.12843}.
	\\
	\item [] Chapter \ref{chap:limitdistribution}
	\\
	\subitem 5. P. Almir\'on, M. Schulze: \textbf{Limit spectral distribution for non-degenerate hypersurface singularities}. To appear in Comptes Rendus Mathématique. \href{https://arxiv.org/pdf/2012.06360.pdf}{arxiv: 2012.06360}.
	\\
	\item []Chapter \ref{chap:wilffunc}
	\\
	\subitem 6. P. Almirón, J.J. Moyano-Fernández: \textbf{Eliahou number, Wilf function and concentration of a numerical semigroup}. To appear in Quaestiones Mathematicae. \href{https://arxiv.org/abs/2104.03793.pdf}{Arxiv: 2104.03793}.
	\\
	\item []Chapter \ref{chap:wilfextension}
	\\	
	\subitem 7. P. Almirón, J.J. Moyano-Fernández: \textbf{A formula for the conductor of a semimodule of a numerical semigroup with two generators}. Semigroup Forum \textbf{103}, 278–285 (2021). \href{https://doi.org/10.1007/s00233-021-10182-1}{DOI:10.1007/s00233-021-10182-1}.
	
	\subitem 8.	P. Almirón, J.J. Moyano-Fernández: \textbf{Supersymmetric gaps of a numerical semigroup with two generators}.\href{https://arxiv.org/pdf/2011.01690.pdf}{Arxiv:2011.01690}.
	
	\subitem 9. P. Almirón, J.J. Moyano-Fernández: \textbf{An Extension of the Wilf conjecture to semimodules over a numerical semigroup}.\href{https://arxiv.org/pdf/2012.01358.pdf}{Arxiv:2012.01358}.
	\\
	\item [] Chapter \ref{chap:torsion}
	\\
	\subitem 10. M.Alberich-Carramiñana, P. Almirón, J.J. Moyano-Fernández: \textbf{Curve singularities with one Puiseux pair and value sets of modules over their local rings}.\href{https://arxiv.org/pdf/2105.07943.pdf}{Arxiv:2105.07943}.
	
\end{enumerate}




%%% Please write your preface here
%Recuerdo los comienzos del año 2011 en los que tenía decidido dedicarme a la música y nos hablaron en el instituto de unos Campus Científicos de Verano financiados por FECYT y el ministerio correspondiente. Los requisitos para poder asistir: tener un buen expediente y hacer un examen, en el que contamos con la inestimable ayuda de nuestros profesores de instituto. Una beca de dos semanas de introducción a la ciencia en la que todo era "gratis" excepto el traslado a donde se celebraba el campus, en mi caso elegí la Universidad Complutense de Madrid. Después de esa experiencia cambié completamente el rumbo que tenía decidido y decidí hacer la carrera de matemáticas en la Universidad Complutense de Madrid. Según google maps la ruta más rápida de la puerta de mi casa en Cazorla a la puerta de la facultad de Ciencias Matemáticas de la UCM es de 363 km, obviamente en coche.
%
%Una beca de excelencia por mi expediente académico, la beca general del ministerio de educación y el esfuerzo de mis padres permitieron que pudiera realizar el primer año de carrera. Recuerdo los nervios de los exámenes de cada cuatrimestre durante cuatro años con la presión de tener que aprobar más de 3/4 de las asignaturas para volver a poder tener la beca general del ministerio. Sin ella, mi permanencia en Madrid se hubiera antojado muy difícil. Cuatro años disfruté de ella, lo que me permitió sufragar la mayoría de los gastos de alojamiento en Madrid. Después de esto, decidí hacer el máster en matemática avanzada en la Universidad Politécnica de Catalunya. Nuevamente, esto fue posible gracias a las becas del ministerio de educación para realizar máster. Y después del máster....viene el doctorado. 
%
%Nuevamente necesitaba una beca sin la que, sin duda, nunca hubiera podido realizar este trabajo. Un año tardó en resolverse. Pero la beca llegó. He disfrutado de 4 años de un contrato estable bajo el paraguas de un grupo de investigación, que además me ha permitido viajar por muchas partes del mundo: Alemania, Hungría, Francia o México.
%
%Son tiempos en donde la sanidad pública y la educación pública se están viendo recortadas en España. No habría nada peor que olvidar de dónde venimos y gracias a qué. Estar orgullosos y defender que vivimos en un país donde gracias a lo público muchos de nosotros tenemos acceso a una educación y a una sanidad de una calidad que en muy pocos países podríamos tener sin endeudarnos hasta las cejas. A pesar de esto mucha gente no puede llegar a tener una carrera o a tener el tratamiento médico que necesita. 
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\chapter{Un capítulo}\label{chap:1}

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% to alter or adjust the chapter heading in the running head

%\abstract*{Each chapter should be preceded by an abstract (10--15 lines long) that summarizes the content. The abstract will appear \textit{online} at \url{www.SpringerLink.com} and be available with unrestricted access. This allows unregistered users to read the abstract as a teaser for the complete chapter. As a general rule the abstracts will not appear in the printed version of your book unless it is the style of your particular book or that of the series to which your book belongs.
%Please use the 'starred' version of the new Springer \texttt{abstract} command for typesetting the text of the online abstracts (cf. source file of this chapter template \texttt{abstract}) and include them with the source files of your manuscript. Use the plain \texttt{abstract} command if the abstract is also to appear in the printed version of the book.}

%\abstract{Each chapter should be preceded by an abstract (10--15 lines long) that summarizes the content. The abstract will appear \textit{online} at \url{www.SpringerLink.com} and be available with unrestricted access. This allows unregistered users to read the abstract as a teaser for the complete chapter. As a general rule the abstracts will not appear in the printed version of your book unless it is the style of your particular book or that of the series to which your book belongs.\newline\indent}
%Please use the 'starred' version of the new Springer \texttt{abstract} command for typesetting the text of the online abstracts (cf. source file of this chapter template \texttt{abstract}) and include them with the source files of your manuscript. Use the plain \texttt{abstract} command if the abstract is also to appear in the printed version of the book.}
\section{Nociones preliminares}

En el capítulo \ref{chap:1} estudiamos la obra \cite{zariski}.

71 KB — too long to show here. It arrives with the project.

\begin{center}

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{\Large \textbf {Universidad Complutense de Madrid }}

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{\Large Facultad de Ciencias Matem\'aticas}

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\includegraphics[width=0.2\textwidth]{{UCM.eps}}

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{\Large \textbf{TESIS DOCTORAL}} 


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 {\Huge 
 
  Title of the thesis. 
  
  Título de la tesis 
  
  }
 
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\bigskip 
		

{\large Memoria para optar al grado de doctor presentada por }


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  {\LARGE                Nombre y apellidos completo del doctorando/a  }
  
  
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\bigskip 
 
{\large Director/a/es: }


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{\Large Nombre y apellidos completos de los directores}		


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{\large Año} 

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