IJPAM: Volume 26, No. 3 (2006)

SMOOTH CURVES COVERING TWO CURVES
OF POSITIVE GENUS

E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it


Abstract.Fix integers $z \ge 2$, $q_i \ge 0$, $1 \le i \le z$, $a_i \ge 2$, $1 \le i \le z$, $g_j$, $1 \le j \le z-1$. Set $k_z:= a_z$ and define by decreasing induction the integers $k_j$, $1 \le j \le z-1$ by the formula $k_j:= a_jk_{j+1}$. Fix smooth and connected projective curves $Y_i$, $1 \le i \le z$, such that $p_a(Y_i)=q_i$ and there is $M_j\in \mbox{Pic}^{a_j}(Y_j)$, $1 \le j \le z$, spanned by its global sections. Set $x_j:= h^0(Y_j,M_j)$ and $g:= g_z$. Assume $1+a_1a_2+a_1q_1+a_2q_2-a_1-a_2 +1-\lfloor x_1/2\rfloor \cdot \lfloor x_2/2\rfloor \le g_1 \le 1+a_1a_2+a_1q_1+a-2q_2-a_1-a_2$ and $1+a_ja_{j+1}+a_jg_j+a_{j+1}q_{j+1}-a_j-a_{j+1} +1-\lfloor x_j/2\rfloor \cdot \l...
...r x_{j+1}/2\rfloor \le g_1 \le 1+a_ja_{j+1}
+a_jg_j+a_{j+1}q_{j+1}-a_j-a_{j+1} $ for $1 \le j \le z-1$. Here we prove the existence of a smooth and connected genus $g_z$ projective curve $X$ and degree $k_i$ morphisms $f_i: X \to Y_i$, $1 \le i \le z$, such that $p_a(X) = g$, $p_a(Y_i)=q_i$, and all morphisms $(f_j,f_z): X \to Y_j\times Y_z$, $1 \le i \le
z-1$, are birational onto its image.

Received: October 21, 2005

AMS Subject Classification: 14H30

Key Words and Phrases: multiple coverings of curves

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 26
Issue: 3