Andy Neitzke
Apr15-04, 07:31 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Robert C. Helling wrote:\n\n> This problem only arises if I try to get the curve as the image of\n> embedding a CP1 into CP3 via a quadratic map (as the Santa Barbara\n> group does).\n\nI am not sure that I 100% understood what you are saying, but I think you\nare discussing the issue of whether the degenerate curves can arise as maps\nCP^1 --> CP^{3|4}. Indeed, as you noted, they do _not_ arise as such maps,\nso if we realize the set of nondegenerate curves as the GL_2 quotient of\nthis map space (which we want to do, because that\'s the description in\nwhich we know the measure) then we have to compactify it if we want to\ndescribe the degenerate ones. One convenient compactification is given by\nthis "space of stable maps." The computations in sections 3 and 4 amount\nto choosing a particular coordinate system near a boundary divisor in this\nspace of stable maps, and studying the behavior of the integrand near that\nboundary divisor.\n\nIt\'s true that in some other descriptions, where you describe the curve by\nequations it satisfies inside the projective space, the\ncompactification looks sort of "automatic" -- to get the singular curve you\njust have to allow some discriminant to vanish. But as Lubos remarked, the\nmost general curve of degree d doesn\'t have this kind of description.\n\n--\nAndy Neitzke\nneitzke@fas.harvard.edu\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Robert C. Helling wrote:
> This problem only arises if I try to get the curve as the image of
> embedding a CP1 into CP3 via a quadratic map (as the Santa Barbara
> group does).
I am not sure that I 100% understood what you are saying, but I think you
are discussing the issue of whether the degenerate curves can arise as maps
CP^1 --> CP^{3|4}. Indeed, as you noted, they do _not_ arise as such maps,
so if we realize the set of nondegenerate curves as the GL_2 quotient of
this map space (which we want to do, because that's the description in
which we know the measure) then we have to compactify it if we want to
describe the degenerate ones. One convenient compactification is given by
this "space of stable maps." The computations in sections 3 and 4 amount
to choosing a particular coordinate system near a boundary divisor in this
space of stable maps, and studying the behavior of the integrand near that
boundary divisor.
It's true that in some other descriptions, where you describe the curve by
equations it satisfies inside the projective space, the
compactification looks sort of "automatic" -- to get the singular curve you
just have to allow some discriminant to vanish. But as Lubos remarked, the
most general curve of degree d doesn't have this kind of description.
--
Andy Neitzke
neitzke@fas.harvard.edu
> This problem only arises if I try to get the curve as the image of
> embedding a CP1 into CP3 via a quadratic map (as the Santa Barbara
> group does).
I am not sure that I 100% understood what you are saying, but I think you
are discussing the issue of whether the degenerate curves can arise as maps
CP^1 --> CP^{3|4}. Indeed, as you noted, they do _not_ arise as such maps,
so if we realize the set of nondegenerate curves as the GL_2 quotient of
this map space (which we want to do, because that's the description in
which we know the measure) then we have to compactify it if we want to
describe the degenerate ones. One convenient compactification is given by
this "space of stable maps." The computations in sections 3 and 4 amount
to choosing a particular coordinate system near a boundary divisor in this
space of stable maps, and studying the behavior of the integrand near that
boundary divisor.
It's true that in some other descriptions, where you describe the curve by
equations it satisfies inside the projective space, the
compactification looks sort of "automatic" -- to get the singular curve you
just have to allow some discriminant to vanish. But as Lubos remarked, the
most general curve of degree d doesn't have this kind of description.
--
Andy Neitzke
neitzke@fas.harvard.edu