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UID:news757@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20190212T003452
DTSTART;TZID=Europe/Zurich:20120511T103000
SUMMARY:Seminar Algebra and Geometry: Rosa M. Miró-Roig (Universitat de Ba
 rcelona)
DESCRIPTION:In my talk\, I will construct families of non-isomorphic Arithm
 etically  Cohen Macaulay (ACM) sheaves (i.e.\, sheaves without intermediat
 e  cohomology) on projective varieties. Since the seminal result by  Horro
 cks characterizing ACM bundles on $\\mathbb{P}^n$ as those that  split int
 o a sum of line bundles\, an important amount of research has  been devote
 d to the study of ACM sheaves on a given variety.  ACM  sheaves also prov
 ide a criterium to determine the complexity of the  underlying variety. Mo
 re concretely\, this complexity can be studied in  terms of the dimension 
 and number of families of indecomposable ACM  sheaves that it supports\, n
 amely its \\emph{representation type}. Along  this line\, a variety that a
 dmits only a finite number of indecomposable  ACM sheaves (up to twist and
  isomorphism) is called of finite  representation type. These varieties ar
 e completely classified: They  are either three or less reduced points in 
 $\\mathbb{P}^2$\, a projective  space $\\mathbb{P}_k^n$\, a smooth quadric
  hypersurface  $X\\subset\\mathbb{P}^n$\, a cubic scroll in $\\mathbb{P}_k
 ^4$\, the Veronese  surface in $\\mathbb{P}_k^5$ or a rational normal curv
 e.  On  the other extreme of complexity\, we would find the varieties of 
 wild representation type\, namely\, varieties for which there exist r-dime
 nsional families of non-isomorphic indecomposable ACM sheaves  for arbitra
 ry large r. In the case of dimension one\, it is known that  curves of wil
 d representation type are exactly those of genus larger or  equal than two
 . In dimension greater or equal than two few examples are  known ans in ma
 y talk\, I will give a brief account of the known results.
X-ALT-DESC: In my talk\, I will construct families of non-isomorphic Arithm
 etically  Cohen Macaulay (ACM) sheaves (i.e.\, sheaves without intermediat
 e  cohomology) on projective varieties. Since the seminal result by  Horro
 cks characterizing ACM bundles on $\\mathbb{P}^n$ as those that  split int
 o a sum of line bundles\, an important amount of research has  been devote
 d to the study of ACM sheaves on a given variety. <br />&nbsp\;<br />ACM  
 sheaves also provide a criterium to determine the complexity of the  under
 lying variety. More concretely\, this complexity can be studied in  terms 
 of the dimension and number of families of indecomposable ACM  sheaves tha
 t it supports\, namely its \\emph{representation type}. Along  this line\,
  a variety that admits only a finite number of indecomposable  ACM sheaves
  (up to twist and isomorphism) is called of <i>finite  representation type
 </i>. These varieties are completely classified: They  are either three or
  less reduced points in $\\mathbb{P}^2$\, a projective  space $\\mathbb{P}
 _k^n$\, a smooth quadric hypersurface  $X\\subset\\mathbb{P}^n$\, a cubic 
 scroll in $\\mathbb{P}_k^4$\, the Veronese  surface in $\\mathbb{P}_k^5$ o
 r a rational normal curve. <br />&nbsp\;<br />On  the other extreme of com
 plexity\, we would find the varieties of <i>wild representation type</i>\,
  namely\, varieties for which there exist r-dimensional families of non-is
 omorphic indecomposable ACM sheaves  for arbitrary large r. In the case of
  dimension one\, it is known that  curves of wild representation type are 
 exactly those of genus larger or  equal than two. In dimension greater or 
 equal than two few examples are  known ans in may talk\, I will give a bri
 ef account of the known results.
DTEND;TZID=Europe/Zurich:20120511T120000
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