Preprint serie C213/2017
Effective divisors in Mg generated by wronskian classes in Mg,2
Gabriel Armando Muñoz Márquez
Keywords: moduli space of curves, effective divisors, limit linear series

This work addresses the problem of computing the class of a certain effective divisor of the moduli space of stable curves of genus g. This divisor is defined as the closure of the locus of smooth curves C having a pair of points (P,Q) with Q having ramification weight at least 3 in the linear system of global sections of omega_{C}(-P).  
Our approach is to combine the methods by Cumino, Esteves and Gatto and the method of test curves. Writing the class of the divisor we want to compute as a linear combination: a*lambda-a_{0}*delta_{0}-a_{1}*delta_{1}-...-a_{[g/2]}*delta_{[g/2]}, we obtain the coefficient a_{i} in terms of the coefficient a_{1} for every i>1 and each odd integer g>=5. We find the following relations:
a_i=(i(g-i)/(g-1))*a_1, for every 2<=i<= [g/2].
Also, we compute the coefficient 'a' by using the Thom--Porteous formula and intersection theory. For each g, we get
a=9g^5-51g^4+129g^3-207g^2+174g-54.

Finally, under a certain hypothesis and using the methods by Cumino, Esteves and Gatto, we find the following inequalities for each g:
-b_i<=a_i for every 1<= i<= [g/2],
where
b_i:=6i^4g^2-6i^4g+12i^4-6i^3g^3-3i^3g^2-3i^3g-18i^3+3i^2g^4+3i^2g^2+12i^2g+6i^2-3ig^5+12ig^4-21ig^3+21ig^2-21ig+6i.
 

 

 

MSC 2000: Algebraic Geometry
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