On the space of morphisms into generic real algebraic varieties

Ghiloni, Riccardo

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 3, pp. 419-438

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 ${\left\{D_i\right\}}_{i=1}^n$ of irreducible real algebraic curves with genus $\ge 2$ and a biregular embedding of $Z$ into the product variety ${\prod }_{i=1}^n{D}_i$. A bijective map $\varphi :{\tilde{Z}}^{\phantom{1}}\to Z$ from a real algebraic variety $\tilde{Z}$ 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 ${\{{\varphi }_t:{\tilde{Z}}_t\to Z\}}_{t\in {ℝ}^k}$ of weak changes of the algebraic structure of $Z$ such that ${\tilde{Z}}_0=Z$, ${\varphi }_0$ is the identity map on $Z$ and, for each $t\in {ℝ}^k\setminus \left\{0\right\}$, ${\tilde{Z}}_t$ 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 $C^{\infty }$ 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.

Classification : 14P05, 14P20

[1] A. Alzati and G. P. Pirola, Some remarks on the de Franchis theorem, Ann. Univ. Ferrara 36 (1990), 45-52. | Zbl | MR

[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. | Zbl | MR

[3] R. Benedetti and A. Tognoli, On real algebraic vector bundles, Bull. Sci. Math. 104 (1980), 89-112. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[6] J. Bochnak and W. Kucharz, The Weierstrass approximation theorem for maps between real algebraic varieties, Math. Ann. 314 (1999), 601-612. | Zbl | MR

[7] J. Bochnak and W. Kucharz, Smooth maps and real algebraic morphisms, Canad. Math. Bull. 42 (1999), 445-451. | Zbl | MR

[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. | Zbl | MR

[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). | Zbl | MR | Numdam

[10] M. Coste and M. Shiota, Nash triviality in families of Nash manifolds, Invent. Math. 108 (1992), 349-368. | Zbl | MR

[11] M. De Franchis, Un teorema sulle involuzioni irrazionali, Rend. Circ. Mat. Palermo 36 (1913), 368. | JFM

[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). | Zbl | MR

[13] R. Ghiloni, Rigidity and moduli space in Real Algebraic Geometry, Math. Ann. 335 (2006), 751-766. | Zbl | MR

[14] M. W. Hirsch, “Differential Topology”, Springer-Verlag, Berlin, 1976. | Zbl | MR

[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. | Zbl | MR | Numdam

[16] N. Joglar-Prieto, Rational surfaces and regular maps into the 2-dimensional sphere, Math. Z. 234 (2000), 399-405. | Zbl | MR

[17] N. Joglar-Prieto and J. Kollár, Real abelian varieties with many line bundles, Bull. London Math. Soc. 35 (2003), 79-84. | Zbl | MR

[18] N. Joglar-Prieto and F. Mangolte, Real algebraic morphisms and Del Pezzo surfaces of degree 2, J. Algebraic Geom. 13 (2004), 269-285. | Zbl | MR

[19] E. Kani, Bounds on the number of non-rational subfields of a function field, Invent. Math. 85 (1986), 185-198. | Zbl | MR

[20] W. Kucharz, Algebraic morphisms into rational real algebraic surfaces, J. Algebraic Geom. 8 (1999), 569-579. | Zbl | MR

[21] W. Kucharz and K. Rusek, Approximation of smooth maps by real algebraic morphisms, Canad. Math. Bull. 40 (1997), 456-463. | Zbl | MR

[22] F. Mangolte, Real algebraic morphisms on 2-dimensional conic bundles, Adv. Geom., to appear. | Zbl | MR

[23] H. Martens, Observations on morphisms of closed Riemann surfaces, Bull. London Math. Soc. 10 (1978), 209-212. | Zbl | MR

[24] J. C. Naranjo and G. P. Pirola, Bounds of the number of rational maps between varieties of general type, to appear. | Zbl | MR

[25] J.-P. Serre, Faisceaux algébriques cohérents, (French), Ann. Math. 61 (1955), 197-278. | Zbl | MR

[26] F. Severi, Sugli integrali abeliani riducibili, Rend. Mat. Acc. Lincei Ser. V, 23 (1914), 581-587. | JFM

[27] M. Shiota, “Nash Manifolds”, Lecture Notes in Mathematics, Vol. 1269, Springer-Verlag, Berlin, 1987. | Zbl | MR

[28] M. Tanabe, A bound for the theorem of de Franchis, Proc. Amer. Math. Soc. 127 (1999), 2289-2295. | Zbl | MR