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UID:news760@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20190212T004214
DTSTART;TZID=Europe/Zurich:20120413T103000
SUMMARY:Seminar Algebra and Geometry: Kevin Langlois (Institut Fourier\, Gr
 enoble)
DESCRIPTION:Let A=C[f1\,...\, fr] be an integral algebra of finite type ove
 r the field of complex numbers.  Using the elements f1\,..\,fr it is diff
 icult in general to  describe the normalization of A. In this talk\, we p
 rovide some examples whenever A is a multigraded algebra. Consider the gro
 up  T=C*×...×C*=(C*)n given by the componentwise multiplication. We say
  that T is an algebraic torus of dimension n. Let M be the character latti
 ce of T. Then a T-action on X=Spec A is equivalent to endow A with a M-gra
 duation. We classify the M-graded algebras A by a number called complexity
 .  Geometrically\, it corresponds to the codimension of general T-orbits
   in X. Algebraically\, the complexity is somehow "the thickness of  gra
 ded pieces" of the algebra A. The  problem of normalization for complexity
  zero case is well known  (monomial or toric case).  For the complexity o
 ne\, the  normalization of A  admits a construction due  to Timashev and
  Altmann-Hausen in terms of  polyhedral divisors over an algebraic  smoot
 h curve. Taking homogeneous  generators\,  we will explain how to build t
 he polyhedral divisor  corresponding  to the normalization of A.  Assume
  that A is normal. Then A  is given by  a polyhedral divisor. A similar p
 roblem arises for the  integral  closure of homogeneous ideals. We will g
 ive an answer for the complexity one case. We will provide also a classifi
 cation of homogeneous integrally  closed ideals of A.
X-ALT-DESC: Let <i>A</i>=<b>C</b>[f<sub>1</sub>\,...\, f<sub>r</sub>] be an
  integral algebra of finite type over the field of complex numbers.&nbsp\;
  Using the elements f<sub>1</sub>\,..\,f<sub>r</sub> it is difficult in ge
 neral to&nbsp\; describe the normalization of<i> A</i>. <br /><br />In thi
 s talk\, we provide some examples whenever <i>A</i> is a multigraded algeb
 ra. <br /><br />Consider the group&nbsp\; <b>T</b>=<b>C</b><sup>*</sup>×.
 ..×<b>C</b><sup>*</sup>=(<b>C</b><sup>*</sup>)<sup>n</sup> given by the c
 omponentwise multiplication. We say that <b>T </b>is an algebraic torus of
  dimension n. Let <i>M</i> be the character lattice of <b>T</b>. Then a <b
 >T</b>-action on <i>X</i>=Spec <i>A</i> is equivalent to endow <i>A</i> wi
 th a <i>M</i>-graduation. <br /><br />We classify the<b> </b><i>M</i>-grad
 ed algebras <i>A</i> by a number called complexity.&nbsp\; Geometrically\,
  it corresponds to the codimension of general <b>T</b>-orbits&nbsp\; in <i
 >X</i>. Algebraically\, the complexity is somehow &quot\;the thickness of&
 nbsp\; graded pieces&quot\; of the algebra <i>A</i>. <br /><br />The  prob
 lem of normalization for complexity zero case is well known  (monomial or 
 toric case).&nbsp\; For the complexity one\, the&nbsp\; normalization of <
 i>A</i>  admits a construction due&nbsp\; to Timashev and Altmann-Hausen i
 n terms of  polyhedral divisors over an algebraic&nbsp\; smooth curve. Tak
 ing homogeneous  generators\,&nbsp\; we will explain how to build the poly
 hedral divisor  corresponding&nbsp\; to the normalization of <i>A</i>.&nbs
 p\; <br /><br />Assume that <i>A</i> is normal. Then <i>A</i>  is given by
 &nbsp\; a polyhedral divisor. A similar problem arises for the  integral&n
 bsp\; closure of homogeneous ideals. We will give an answer for the comple
 xity one case. We will provide also a classification of homogeneous integr
 ally&nbsp\; closed ideals of <i>A</i>.
DTEND;TZID=Europe/Zurich:20120413T120000
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