JournalscmhVol. 78, No. 2pp. 227–244

Asymptotic behaviour of Betti numbers of real algebraic surfaces

  • F. Bihan

    Université de Lausanne, Switzerland
Asymptotic behaviour of Betti numbers of real algebraic surfaces cover
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Abstract

Let XmX_m be a nonsingular real algebraic surface of degree m in the complex projective space CP3{\mathbb C}P^3 and RXm{\mathbb R}X_m its real point set in RP3{\mathbb R}P^3. In the spirit of the sixteenth Hilbert's problem, one can ask for each degree m about the maximal possible value βi,m\beta_{i,m} of the Betti number bi(RXm)b_i({\mathbb R}X_m) (i=0 or 1). We show that βi,m\beta_{i,m} is asymptotically equivalent to lim3l_i \cdot m^3 for some real number lil_i and prove inequalities 1336l0512\frac{13}{36} \leq l_0 \leq \frac{5}{12} and 1318l156\frac{13}{18} \leq l_1 \leq \frac{5}{6}.

Cite this article

F. Bihan, Asymptotic behaviour of Betti numbers of real algebraic surfaces. Comment. Math. Helv. 78 (2003), no. 2, pp. 227–244

DOI 10.1007/S000140300010