Universal relations on stable map spaces in genus zero
Authors:
Anca M. Mustata and Andrei Mustata
Journal:
Trans. Amer. Math. Soc. 362 (2010), 16991720
MSC (2000):
Primary 14N35, 14F25
Published electronically:
October 28, 2009
MathSciNet review:
2574874
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We introduce a factorization for the map between moduli spaces of stable maps which forgets one marked point. This leads to a study of universal relations in the cohomology of stable map spaces in genus zero.
 [BO]
K.
Behrend and A.
O’Halloran, On the cohomology of stable map spaces,
Invent. Math. 154 (2003), no. 2, 385–450. MR 2013785
(2004k:14002), http://dx.doi.org/10.1007/s0022200303085
 [CHS]
I. Coskun, J. Harris, J. M. Starr, The ample cone of the Kontsevich moduli space, preprint.
 [C1]
J. Cox, An additive basis for the cohomology ring of , in math.AG/0501322
 [C2]
J. Cox, A presentation for the cohomology ring in math.AG/0505112
 [FP]
W.
Fulton and R.
Pandharipande, Notes on stable maps and quantum cohomology,
Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math.,
vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45–96.
MR
1492534 (98m:14025)
 [GP]
E. Getzler, R. Pandharipande, The Betti numbers of , in math.AG/0502525
 [Has]
Brendan
Hassett, Moduli spaces of weighted pointed stable curves, Adv.
Math. 173 (2003), no. 2, 316–352. MR 1957831
(2004b:14040), http://dx.doi.org/10.1016/S00018708(02)000580
 [Ka]
M.
M. Kapranov, Chow quotients of Grassmannians. I, I. M.
Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc.,
Providence, RI, 1993, pp. 29–110. MR 1237834
(95g:14053)
 [Ke]
Sean
Keel, Intersection theory of moduli space of
stable 𝑛pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), no. 2, 545–574. MR 1034665
(92f:14003), http://dx.doi.org/10.1090/S00029947199210346650
 [KK]
Alexandre
Kabanov and Takashi
Kimura, A change of coordinates on the large phase space of quantum
cohomology, Comm. Math. Phys. 217 (2001), no. 1,
107–126. MR 1815027
(2002d:14092), http://dx.doi.org/10.1007/s002200000359
 [LP]
Y.P.
Lee and R.
Pandharipande, A reconstruction theorem in quantum cohomology and
quantum 𝐾theory, Amer. J. Math. 126 (2004),
no. 6, 1367–1379. MR 2102400
(2006c:14082)
 [MM1]
Andrei
Mustaţă and Magdalena
Anca Mustaţă, Intermediate moduli spaces of stable
maps, Invent. Math. 167 (2007), no. 1,
47–90. MR
2264804 (2008j:14024), http://dx.doi.org/10.1007/s0022200600061
 [MM2]
A. Mustata, A. Mustata, The Chow ring of , math.AG/0507464
 [MM3]
A. Mustata, A. Mustata, Tautological rings of stable map spaces, preprint.
 [MM4]
A. Mustata, A. Mustata, On Chow quotients, in preparation.
 [O1]
D. Oprea, The tautological rings of the moduli spaces of stable maps, in math.AG/0404280
 [Pan]
Rahul
Pandharipande, Intersections of 𝐐divisors on
Kontsevich’s moduli space
\overline𝐌_{0,𝐧}(𝐏^{𝐫},𝐝) and
enumerative geometry, Trans. Amer. Math.
Soc. 351 (1999), no. 4, 1481–1505. MR 1407707
(99f:14068), http://dx.doi.org/10.1090/S0002994799019091
 [Par]
A. Parker, An Elementary GIT Construction of the Moduli Space of Stable Maps, in math.AG/0604092
 [W]
Edward
Witten, Twodimensional gravity and intersection theory on moduli
space, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh
Univ., Bethlehem, PA, 1991, pp. 243–310. MR 1144529
(93e:32028)
 [BO]
 K. Behrend, A. O'Halloran, On the cohomology of stable map spaces, in Invent. Math. 154 (2003), no. 2, 385450. MR 2013785 (2004k:14002)
 [CHS]
 I. Coskun, J. Harris, J. M. Starr, The ample cone of the Kontsevich moduli space, preprint.
 [C1]
 J. Cox, An additive basis for the cohomology ring of , in math.AG/0501322
 [C2]
 J. Cox, A presentation for the cohomology ring in math.AG/0505112
 [FP]
 W. Fulton, R. Pandharipande, Notes on stable maps and quantum cohomology in Algebraic geometry, Santa Cruz 1995, pages 4596, volume 62 of Proc. Symp. Pure Math., Amer. Math. Soc., 1997. MR 1492534 (98m:14025)
 [GP]
 E. Getzler, R. Pandharipande, The Betti numbers of , in math.AG/0502525
 [Has]
 B. Hassett, Moduli spaces of weighted pointed stable curves. Adv. Math. 173 (2003), no. 2, 316352. MR 1957831 (2004b:14040)
 [Ka]
 M. M. Kapranov, Chow quotients of Grassmannians. I, in I. M. Gelfand Seminar, 29110, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, RI, 1993. MR 1237834 (95g:14053)
 [Ke]
 S. Keel, Intersection theory of moduli space of stable pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), page 545574. MR 1034665 (92f:14003)
 [KK]
 A. Kabanov, T. Kimura, A change of coordinates on the large phase space of quantum cohomology. Comm. Math. Phys. 217 (2001), no. 1, 107126. MR 1815027 (2002d:14092)
 [LP]
 Y.P. Lee, R. Pandharipande, A reconstruction theorem in quantum cohomology and quantum theory, Amer. J. Math. 126 (2004), no. 6, 13671379. MR 2102400 (2006c:14082)
 [MM1]
 A. Mustata, A. Mustata, Intermediate moduli spaces of stable maps, math.AG/0409569, Invent. Math. 167 (2007), no. 1, 4790 MR 2264804 (2008j:14024)
 [MM2]
 A. Mustata, A. Mustata, The Chow ring of , math.AG/0507464
 [MM3]
 A. Mustata, A. Mustata, Tautological rings of stable map spaces, preprint.
 [MM4]
 A. Mustata, A. Mustata, On Chow quotients, in preparation.
 [O1]
 D. Oprea, The tautological rings of the moduli spaces of stable maps, in math.AG/0404280
 [Pan]
 R. Pandharipande, Intersection of divisors on Kontsevich's Moduli Space and enumerative geometry, Trans. Amer. Math. Soc. 351 (1999), no. 4, 14811505. MR 1407707 (99f:14068)
 [Par]
 A. Parker, An Elementary GIT Construction of the Moduli Space of Stable Maps, in math.AG/0604092
 [W]
 E. Witten, Twodimensional gravity and intersection theory on moduli space. Surveys in differential geometry (Cambridge, MA, 1990), 243310, Lehigh Univ., Bethlehem, PA, 1991. MR 1144529 (93e:32028)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
14N35,
14F25
Retrieve articles in all journals
with MSC (2000):
14N35,
14F25
Additional Information
Anca M. Mustata
Affiliation:
School of Mathematical Sciences, 153 Aras Na Laoi, University College Cork, Cork, Ireland
Email:
A.Mustata@ucc.ie
Andrei Mustata
Affiliation:
School of Mathematical Sciences, 153 Aras Na Laoi, University College Cork, Cork, Ireland
DOI:
http://dx.doi.org/10.1090/S0002994709046066
PII:
S 00029947(09)046066
Received by editor(s):
February 1, 2007
Published electronically:
October 28, 2009
Article copyright:
© Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
