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SUMMARY:Andreas Leopold Knutsen (Bergen)
DTSTART:20260424T124000Z
DTEND:20260424T134000Z
DTSTAMP:20260423T034446Z
UID:OBAGS/85
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OBAGS/85/">D
 istinguishing Brill-Noether loci</a>\nby Andreas Leopold Knutsen (Bergen) 
 as part of ODTU-Bilkent Algebraic Geometry Seminars\n\n\nAbstract\nBrill-N
 oether theory has since the end of the 19th century studied linear systems
  on smooth projective curves (or\, equivalently\, compact Riemann surfaces
 ). A degree $d$ linear system\nof dimension $r$ on a curve $C$\, called a 
 $g^r_d$\, roughly corresponds to a non-degenerate morphism $C \\to \\mathb
 b{P}^r$ of degree $d$. In the moduli space $\\mathcal{M}_g$ of curves of g
 enus $g$ one can define the Brill-Noether loci\n\\[ \\mathcal{M}^r_{g\,d}:
 = \\{ C \\in \\mathcal{M}_g \\\; : \\\; C \\\; {\\rm has~a} \\\; g^r_d \\}
  \,  \\]\nwhich are closed subvarieties.\n\nThe classical Brill-Noether-Pe
 tri theorem\, proved first by D. Gieseker in 1982\,\nstates that a general
  smooth curve $C$ of genus $g$ admits a linear system of dimension $r$ and
  degree $d$\nif and only if the Brill–Noether number\n\\[ \\rho(g\, r\, 
 d): = g - (r+ 1)(g-d+r) \\geq 0.\\]\nThis can be restated as $\\mathcal{M}
 ^r_{g\,d}=\\mathcal{M}_g$ if and only if $\\rho(g\, r\, d) \\geq 0$. When 
 $\\rho(g\, r\, d) < 0$ it is known that  the codimension of any component 
 of $\\mathcal{M}^r_{g\,d}$ is at most $-\\rho(g\, r\, d)$.\nIn general sur
 prisingly little is known about the geometry of Brill–Noether loci\nwhen
  $\\rho(g\, r\, d) < 0$\, in particular about the containments between the
 m.\n\nI will describe results from a joint work with Asher Auel and Richar
 d Haburcak (arXiv:2406.19993)\, where we determine all the  maximal Brill-
 Noether loci (wrt containment) in terms of numerical conditions on $g\,r\,
 d$ and show that they are all distinct.\n
LOCATION:https://researchseminars.org/talk/OBAGS/85/
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