moduli of curves - j. harris, i. morrison

Moduli of Curves Joe Harris Ian Morrison Springer [...]... to curves B of genus h ≥ 2 One warning about these variants is in order: the notion of scheme of type X” needs to be handled with caution For example, look at the following types of subschemes of P2 : 1 Plane curves of degree d; 2 Reduced and irreducible plane curves of degree d; 3 Reduced and irreducible plane curves of degree d and geometric genus g; and, 5 Perhaps, more accurately, in view of our... component of some Hilbert scheme To illustrate the application of this law, and as an example of a tangent-space-to-the-Hilbert-scheme calculation, we now wish to recall Mumford’s famous example [118] of a component J of the (restricted) Hilbert scheme of space curves that is everywhere nonreduced This example also serves to justify the somewhat technical D Extrinsic pathologies 19 construction of the... also been calling moduli spaces) as parameter spaces In this sense, the space Mg of smooth curves of genus g is a moduli space while the space Hd,g,r of subcurves of Pr of degree d and (arithmetic) genus g is a parameter space The extrinsic element r in the second case is the gd that maps the abstract curve to Pr and the choice of basis of this linear system that fixes the embedding Of course, this distinction... series on flag curves Inequalities on vanishing sequences The case ρ = 0 Proof of the Gieseker-Petri theorem xiii 266 269 274 274 276 280 6 Geometry of moduli spaces: selected results A Irreducibility of the moduli space of curves B Diaz’ theorem The idea: stratifying the moduli space The proof ... coarse moduli space for curves of genus 1 2) Show that a j-function J on a scheme B arises as the j-function associated to a family of curves of genus 1 only if all the multiplicities of the zero-divisor of J are divisible by 3, and all multiplicities of (J − 1728) are even Using this fact, show that M1 is not a fine moduli space for curves of genus 1 3) Show that the family y 2 − x 3 − t over the punctured... commutative fiber-product diagram D (1.2) ✲ C 1 ϕ ❄ B χ ✲ ❄ M with ϕ : D ✲ B in S(B) and Ψ (ϕ) = χ In sum, every family over B is the pullback of C via a unique map of B to M and we have a perfect dictionary enabling us to translate between information about the geometry of families of our moduli problem and information about the geometry of the moduli space M itself One of the main themes of moduli theory... count the dimension of the family of such curves, then, we reverse this analysis, starting with a conic D, which moves with 8 degrees of freedom The projective space Λ of quartics containing D has dimension 25 An open subset of the 48-dimensional Grassmannian G(1, 25) of pencils in Λ will have base locus the union of D and a curve C not lying on any cubic The dimension of the family J4 of all such C is... Finite generation of and separation by invariants The numerical criterion Stability of plane curves B Stability of Hilbert points of smooth curves The numerical criterion for Hilbert points Gieseker’s criterion Stability of smooth curves C Construction of Mg via the Potential Stability Theorem The plan of the construction... mind are wide-ranging For example, we might take our families to be 1 smooth flat morphisms C ✲ B whose fibers are smooth curves of genus g, or 2 subschemes C in Pr × B, flat over B, whose fibers over B are curves of fixed genus g and degree d, and so on We can loosely consider the elements of S(Spec(C)) as the objects of our moduli problem and the elements of S(B) over other bases as families of such objects... happy consequences for the study of F If ϕ : D ✲ B is any family in (i.e., any element of) S(B), then χ = Ψ (ϕ) is a morphism from B to M Intuitively, (closed) points of M classify the objects of our moduli problem and the map χ sends a (closed) point b of B to the moduli point in M determined by the fiber Db of D over b Going the other way, pulling back the identity map of M itself via Ψ constructs a . Limit linear series: definitions and applications 263 Limit linear series 263 Contents xiii Smoothing limit linear series 266 Limits of canonical series and Weierstrass points . . . 269 E Limit. all the multiplicities of the zero-divisor of J are divisible by 3, and all multiplicities of (J −1728) are even. Using this fact, show that M 1 is not a fine moduli space for curves of genus 1. 3). 192 Finite generation of and separation by invariants . . . 194 The numerical criterion 199 Stability of plane curves 202 B Stability of Hilbert points of smooth curves 206 The numerical criterion