On the space of morphisms into generic real algebraic varieties

Riccardo Ghiloni

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

  • Volume: 5, Issue: 3, page 419-438
  • ISSN: 0391-173X

Abstract

top
We introduce a notion of generic real algebraic variety and we study the space of morphisms into these varieties. Let Z be a real algebraic variety. We say that Z is generic if there exist a finite family { D i } i = 1 n of irreducible real algebraic curves with genus ≥ 2 and a biregular embedding of Z into the product variety ∏ i = 1 n D i . A bijective map ϕ : Z ˜ 1 → Z from a real algebraic variety Z ˜ to Z is called weak change of the algebraic structure of  Z if it is regular and its inverse is a Nash map. Generic real algebraic varieties are “generic” in the sense specified by the following result: For each real algebraic variety Z and for integer k , there exists an algebraic family { ϕ t : Z ˜ t → Z } t ∈ ℝ k of weak changes of the algebraic structure of Z such that Z ˜ 0 = Z , ϕ 0 is the identity map on Z and, for each t ∈ ℝ k ∖ { 0 } , Z ˜ t is generic. Let X and Y be nonsingular irreducible real algebraic varieties. Regard the set ℛ ( X , Y ) of regular maps from X to Y as a subspace of the corresponding set 𝒩 ( X , Y ) of Nash maps, equipped with the C ∞  compact-open topology. We prove that, if Y is generic, then ℛ ( X , Y )  is closed and nowhere dense in 𝒩 ( X , Y ) , and has a semi-algebraic structure. Moreover, the set of dominating regular maps from X to Y is finite. A version of the preceding results in which X and Y can be singular is given also.

How to cite

top

Ghiloni, Riccardo. "On the space of morphisms into generic real algebraic varieties." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.3 (2006): 419-438. <http://eudml.org/doc/242766>.

@article{Ghiloni2006,
abstract = {We introduce a notion of generic real algebraic variety and we study the space of morphisms into these varieties. Let $Z$ be a real algebraic variety. We say that $Z$ is generic if there exist a finite family $\lbrace D_i\rbrace _\{i=1\}^n$ of irreducible real algebraic curves with genus $\ge 2$ and a biregular embedding of $Z$ into the product variety $\prod _\{i=1\}^nD_i$. A bijective map $\varphi \!:\!\widetilde\{Z\}^\{\vphantom\{1\}\} \rightarrow Z$ from a real algebraic variety $\widetilde\{Z\}$ to $Z$ is called weak change of the algebraic structure of $Z$ if it is regular and its inverse is a Nash map. Generic real algebraic varieties are “generic” in the sense specified by the following result: For each real algebraic variety $Z$ and for integer $k$, there exists an algebraic family $\lbrace \varphi _t:\widetilde\{Z\}_t \rightarrow Z\rbrace _\{t \in \mathbb \{R\}^k\}$ of weak changes of the algebraic structure of $Z$ such that $\widetilde\{Z\}_0=Z$, $\varphi _0$ is the identity map on $Z$ and, for each $t \in \mathbb \{R\}^k \setminus \lbrace 0\rbrace $, $\widetilde\{Z\}_t$ is generic. Let $X$ and $Y$ be nonsingular irreducible real algebraic varieties. Regard the set $\mathcal \{R\}(X,Y)$ of regular maps from $X$ to $Y$ as a subspace of the corresponding set $\mathcal \{N\}(X,Y)$ of Nash maps, equipped with the $C^\{\infty \}$ compact-open topology. We prove that, if $Y$ is generic, then $\mathcal \{R\}(X,Y)$ is closed and nowhere dense in $\mathcal \{N\}(X,Y)$, and has a semi-algebraic structure. Moreover, the set of dominating regular maps from $X$ to $Y$ is finite. A version of the preceding results in which $X$ and $Y$ can be singular is given also.},
author = {Ghiloni, Riccardo},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {419-438},
publisher = {Scuola Normale Superiore, Pisa},
title = {On the space of morphisms into generic real algebraic varieties},
url = {http://eudml.org/doc/242766},
volume = {5},
year = {2006},
}

TY - JOUR
AU - Ghiloni, Riccardo
TI - On the space of morphisms into generic real algebraic varieties
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2006
PB - Scuola Normale Superiore, Pisa
VL - 5
IS - 3
SP - 419
EP - 438
AB - We introduce a notion of generic real algebraic variety and we study the space of morphisms into these varieties. Let $Z$ be a real algebraic variety. We say that $Z$ is generic if there exist a finite family $\lbrace D_i\rbrace _{i=1}^n$ of irreducible real algebraic curves with genus $\ge 2$ and a biregular embedding of $Z$ into the product variety $\prod _{i=1}^nD_i$. A bijective map $\varphi \!:\!\widetilde{Z}^{\vphantom{1}} \rightarrow Z$ from a real algebraic variety $\widetilde{Z}$ to $Z$ is called weak change of the algebraic structure of $Z$ if it is regular and its inverse is a Nash map. Generic real algebraic varieties are “generic” in the sense specified by the following result: For each real algebraic variety $Z$ and for integer $k$, there exists an algebraic family $\lbrace \varphi _t:\widetilde{Z}_t \rightarrow Z\rbrace _{t \in \mathbb {R}^k}$ of weak changes of the algebraic structure of $Z$ such that $\widetilde{Z}_0=Z$, $\varphi _0$ is the identity map on $Z$ and, for each $t \in \mathbb {R}^k \setminus \lbrace 0\rbrace $, $\widetilde{Z}_t$ is generic. Let $X$ and $Y$ be nonsingular irreducible real algebraic varieties. Regard the set $\mathcal {R}(X,Y)$ of regular maps from $X$ to $Y$ as a subspace of the corresponding set $\mathcal {N}(X,Y)$ of Nash maps, equipped with the $C^{\infty }$ compact-open topology. We prove that, if $Y$ is generic, then $\mathcal {R}(X,Y)$ is closed and nowhere dense in $\mathcal {N}(X,Y)$, and has a semi-algebraic structure. Moreover, the set of dominating regular maps from $X$ to $Y$ is finite. A version of the preceding results in which $X$ and $Y$ can be singular is given also.
LA - eng
UR - http://eudml.org/doc/242766
ER -

References

top
  1. [1] A. Alzati and G. P. Pirola, Some remarks on the de Franchis theorem, Ann. Univ. Ferrara 36 (1990), 45–52. Zbl0774.14029MR1151481
  2. [2] E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, “Geometry of Algebraic Curves”, Vol. I. Grundlehren Math. Wiss., Vol. 267, Springer-Verlag, New York, 1985. Zbl0559.14017MR770932
  3. [3] R. Benedetti and A. Tognoli, On real algebraic vector bundles, Bull. Sci. Math. 104 (1980), 89–112. Zbl0421.58001MR560747
  4. [4] J. Bochnak, M. Coste and M.-F. Roy, “Real Algebraic Geometry”, Translated from the 1987 French original. Revised by the authors. Ergeb. Math. Grenzgeb. (3), Vol. 36, Springer-Verlag, Berlin, 1998. Zbl0912.14023MR1659509
  5. [5] J. Bochnak and W. Kucharz, The Weierstrass approximation theorem and a characterization of the unit circle, Proc. Amer. Math. Soc. 127 (1999), 1571–1574. Zbl0912.14024MR1653417
  6. [6] J. Bochnak and W. Kucharz, The Weierstrass approximation theorem for maps between real algebraic varieties, Math. Ann. 314 (1999), 601–612. Zbl0945.14032MR1709103
  7. [7] J. Bochnak and W. Kucharz, Smooth maps and real algebraic morphisms, Canad. Math. Bull. 42 (1999), 445–451. Zbl0955.14043MR1727342
  8. [8] J. Bochnak and W. Kucharz, Line bundles, regular mappings and the underlying real algebraic structure of complex algebraic varieties, Math. Ann. 316 (2000), 793–817. Zbl0986.14041MR1758454
  9. [9] J. Bochnak, W. Kucharz and R. Silhol, Morphisms, line bundles and moduli spaces in real algebraic geometry, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 5–65 (1998); Erratum to: “Morphisms, line bundles and moduli spaces in real algebraic geometry” [Inst. Hautes Études Sci. Publ. Math. 86, (1997), 5–65 (1998); MR 99h:14055], Inst. Hautes Études Sci. Publ. Math. 92 (2000), 195 (2001). Zbl0938.14033MR1608561
  10. [10] M. Coste and M. Shiota, Nash triviality in families of Nash manifolds, Invent. Math. 108 (1992), 349–368. Zbl0801.14017MR1161096
  11. [11] M. De Franchis, Un teorema sulle involuzioni irrazionali, Rend. Circ. Mat. Palermo 36 (1913), 368. JFM44.0657.02
  12. [12] R. Ghiloni, On the space of real algebraic morphisms, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 14 (2003), 307–317 (2004). Zbl1172.14341MR2104218
  13. [13] R. Ghiloni, Rigidity and moduli space in Real Algebraic Geometry, Math. Ann. 335 (2006), 751–766. Zbl1098.14045MR2232015
  14. [14] M. W. Hirsch, “Differential Topology”, Springer-Verlag, Berlin, 1976. Zbl0356.57001MR448362
  15. [15] A. Howard and A. J. Sommese, On the theorem of de Franchis, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), 429–436. Zbl0534.14016MR739918
  16. [16] N. Joglar-Prieto, Rational surfaces and regular maps into the 2-dimensional sphere, Math. Z. 234 (2000), 399–405. Zbl0968.14033MR1765888
  17. [17] N. Joglar-Prieto and J. Kollár, Real abelian varieties with many line bundles, Bull. London Math. Soc. 35 (2003), 79–84. Zbl1024.14024MR1934435
  18. [18] N. Joglar-Prieto and F. Mangolte, Real algebraic morphisms and Del Pezzo surfaces of degree 2, J. Algebraic Geom. 13 (2004), 269–285. Zbl1067.14059MR2047699
  19. [19] E. Kani, Bounds on the number of non-rational subfields of a function field, Invent. Math. 85 (1986), 185–198. Zbl0615.12017MR842053
  20. [20] W. Kucharz, Algebraic morphisms into rational real algebraic surfaces, J. Algebraic Geom. 8 (1999), 569–579. Zbl0973.14030MR1689358
  21. [21] W. Kucharz and K. Rusek, Approximation of smooth maps by real algebraic morphisms, Canad. Math. Bull. 40 (1997), 456–463. Zbl0918.14022MR1611343
  22. [22] F. Mangolte, Real algebraic morphisms on 2-dimensional conic bundles, Adv. Geom., to appear. Zbl1097.14045MR2243296
  23. [23] H. Martens, Observations on morphisms of closed Riemann surfaces, Bull. London Math. Soc. 10 (1978), 209–212. Zbl0384.14008MR480985
  24. [24] J. C. Naranjo and G. P. Pirola, Bounds of the number of rational maps between varieties of general type, to appear. Zbl1210.14014MR2369893
  25. [25] J.-P. Serre, Faisceaux algébriques cohérents, (French), Ann. Math. 61 (1955), 197–278. Zbl0067.16201MR68874
  26. [26] F. Severi, Sugli integrali abeliani riducibili, Rend. Mat. Acc. Lincei Ser. V, 23 (1914), 581–587. Zbl45.0698.01JFM45.0698.01
  27. [27] M. Shiota, “Nash Manifolds”, Lecture Notes in Mathematics, Vol. 1269, Springer-Verlag, Berlin, 1987. Zbl0629.58002MR904479
  28. [28] M. Tanabe, A bound for the theorem of de Franchis, Proc. Amer. Math. Soc. 127 (1999), 2289–2295. Zbl0919.30035MR1600153

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.