Regular components of moduli spaces of stable maps

نویسنده

  • GAVRIL FARKAS
چکیده

The purpose of this note is to prove the existence of ‘nice’ components of the Hilbert scheme of curves C ⊆ P × P of genus g and bidegree (k, d). We can also phrase our result using the Kontsevich moduli space of stable maps to P × P. For a smooth projective variety Y and a class β ∈ H2(Y,Z), one considers the moduli space Mg(Y, β) of stable maps f : C → Y , with C a reduced connected nodal curve of genus g and f∗([C]) = β (see [FP] for the construction of these moduli spaces). We denote by π : Mg(Y, β) → Mg the natural projection. The expected dimension of the stack Mg(Y, β) is χ(g, Y, β) = dim(Y ) (1− g) + 3g − 3− β ·KY . Since in general the geometry of Mg(Y, β) is quite messy (existence of many components, some nonreduced and/or not of expected dimension), it is not obvious what should be the definition of a good component of Mg(Y, β). Following Sernesi [Se] we introduce the following: Definition: A component V of Mg(Y, β) is said to be regular if it is generically smooth and of dimension χ(g, Y, β). We say that V has the expected number of moduli if

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تاریخ انتشار 2000