Moduli spaces of weighted pointed stable curves

By Brendan Hassett. 2002

363 citations recorded by the source corpus.

Abstract A weighted pointed curve consists of a nodal curve and a sequence of marked smooth points, each assigned a number between zero and one. A subset of the marked points may coincide if the sum of the corresponding weights is no greater than one. We construct moduli spaces for these objects using methods of the log minimal model program, and describe the induced birational morphisms between moduli spaces as the weights are varied. In the genus zero case, we explain the connection to Geometric Invariant Theory quotients of points in the projective line, and to compactifications of moduli spaces studied by Kapranov, Keel, and Losev-Manin.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/410a4c8294d6a954523931edcf3c60e6e96297cb

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Citations in the corpus

K-stability of Fano varieties: an algebro-geometric approach (2020, result): “e Remark 10.7. 10.2. Wall crossings and log surfaces. One interesting new direction is to consider the moduli space of log Fano varieties (X,t∆), and vary the coeﬃcient t. This reminisces the work in [Has03] which studies a similar setting for KSB stable log curves (C,tD). In [ADL19], the authors de”
Conformal blocks and rational normal curves (2010, background): “nt to the image carrying ≤ βj distinct points. Thus the condition P 4 1βk = d+ 2 translates to the image of the entire curve carrying ≤ d+2 distinct points, as claimed. 4.4. Hassett spaces. Following [Has03], for A := {a 1,...,an} ∈ Qn ∩(0,1]n such that P ai ≥ 2 we denote by M 0,A the compact moduli”
Zariski decompositions of numerical cycle classes (2013, methodology): “ollowing diagram: M 0;7 ! M S 7 ? 0;7 ? y ? ? y˚ M 0;A ! M S 7 0;A Here M 0;A= (P1)7==SL 2 is the moduli space of stable weighted pointed curves dened by assigning the weight 1=3 to each point as in [Has03]. The quotient from M 0;7 to M 0;Ais S 7-equivariant and descends to dene the map ˚. The nef d”
Threefold thresholds (2003, result): “In yet another direction, the log canonical threshold should be closely related to recent work of Hassett, [11].”
Alternate Compactifications of Moduli Spaces of Curves (2010, background): “-pointed curve (C;fpign i=1) is A-stable if it satises (1) The only singularities of Care nodes, (2) If pi 1 ;:::;pi k coincide in C, then Pk j=1 aij 1, (3) !C( n i=1aipi) is ample. Corollary 2.14 ([Has03]). There exist proper moduli stacks Mg;Afor A-stable curves, with corresponding moduli spaces M”
Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension (2015, background): “.9). In any case, one should mention that the introduction of fractional coeﬃcients for higher dimensional pairs led Hassett to go back to the case of n-pointed curves and study a weighted version in [Has03]. These moduli spaces are more numerous and have greater ﬂexibility than the traditional ones”
Semipositivity theorems for moduli problems (2012, background): “we do not discuss the moduli spaces of stable pairs here, Theorem 1.12 and Theorem 1.13 play important roles in the proof of the projectivity of the moduli spaces of stable pairs (see [Ale2], [FulP], [Has], and 4.4 below). Theorem 1.13 (Semipositivity theorem III). Let Xbe an equidimensional variety”
MODULI SPACES OF RATIONAL WEIGHTED STABLE CURVES AND TROPICAL GEOMETRY (2014, methodology): “Hassett [H] introduces these spaces in the context of the log minimal model program and proves that there is a smooth projective scheme representing the moduli problem of weighted stable curves, denoted M0,w.”
The moduli space of stable quotients (2009, background): “SMg;mjd constructed in a more general context by Hassett [14].”
Log canonical models for the moduli space of curves: The first divisorial contraction (2006, background): “These can be understood explicitly in special cases, e.g., (M0,n, δ) is a log canonical model [ Has03 ] 7.1.”
Square-tiled surfaces and rigid curves on moduli spaces (2010, background): “ent Dk for any k > 2. The numerical class of T hyp d,µ,i spans an extremal ray of the cone of moving curves on Hg ∼= Mf0,2g+2, which is dual to the face hD3,...,Dg+1i of the eﬀective cone. Hassett [H] studied moduli spaces of weighted pointed stable curves. The moduli space M0,n can be regarded a”
Hypertrees, projections, and moduli of stable rational curves (2010, background): “ birational contraction of M 0;nwhose exceptional locus intersects the interior M 0;n. Birational contractions whose exceptional locus lies in the boundary have been previously constructed by Hassett [Has]. In particular, the map vgives a (hypothetical) new Mori chamber of M 0;n. It would be interes”