Rational maps between moduli spaces of curves and Gieseker-Petri divisors
Rational maps between moduli spaces of curves and Gieseker-Petri divisors
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曲线模空间和 Gieseker-Petri 除数之间的有理映射
DOI：
10.1090/s1056-3911-09-00510-4
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发表时间：
2007
影响因子：
1.8
通讯作者：
G. Farkas
中科院分区：
文献类型：
--
作者：
G. Farkas
是单射的。该定理由 Petri 猜想并由 Gieseker [G] 证明（有关更简化的证明，请参见 [EH3]），是代数曲线理论的基石。这意味着度数为 d、维度为 r 的线性级数的变体 Gd(C) = {(L, V ) : L ∈ Pic (C), V ∈ G(r + 1,H0(L))} 是平滑的且具有预期维度 ρ(g, r, d) := g− (r+1)(g−d+ r) 并且健忘映射 Gd(C) → W r d (C) 是奇点的有理解析（参见[ACGH] 对于许多其他应用）。描述由曲线 [C] ε Mg 组成的轨迹 GPg ⊂ Mg 使得 C 上存在线丛 L，而 Gieseker-Petri 定理失败，这是一个古老的开放问题。显然，GPg 根据线性级数的数值类型分解为不可约分量。对于固定整数 d, r ≥ 1 使得 g − d + r ≥ 2，我们定义由曲线 [C] ∈ Mg 组成的轨迹 GPg,d ，使得存在一对线性级数 (L, V ) ∈ G r d(C) 和 (KC ⊗ L ∨,W ) ∈ G 2g−2−d (C) ，其乘法映射 μ0(V,W ) : V ⊗W → H (C,KC )
is injective. The theorem, conjectured by Petri and proved by Gieseker [G] (see [EH3] for a much simplified proof), lies at the cornerstone of the theory of algebraic curves. It implies that the variety Gd(C) = {(L, V ) : L ∈ Pic (C), V ∈ G(r + 1,H0(L))} of linear series of degree d and dimension r is smooth and of expected dimension ρ(g, r, d) := g− (r+1)(g−d+ r) and that the forgetful mapGd(C) → W r d (C) is a rational resolution of singularities (see [ACGH] for many other applications). It is an old open problem to describe the locus GPg ⊂ Mg consisting of curves [C] ∈ Mg such that there exists a line bundle L on C for which the Gieseker-Petri theorem fails. Obviously GPg breaks up into irreducible components depending on the numerical types of linear series. For fixed integers d, r ≥ 1 such that g − d + r ≥ 2, we define the locus GPg,d consisting of curves [C] ∈ Mg such that there exist a pair of linear series (L, V ) ∈ G r d(C) and (KC ⊗ L ∨,W ) ∈ G 2g−2−d (C) for which the multiplication map μ0(V,W ) : V ⊗W → H (C,KC )
Prym 品种模空间的 Kodaira 维数
影响因子：
2.6
作者：
Farkas;Gavril;Ludwig;Katharina
通讯作者：
Katharina