ACADEMIC YEAR 2012/2013

TALKS

Venerdi 15 Marzo 2013, ore 11:00-13:00 : The log minimal model program for M_g, I (Angelo Lopez)

Venerdi 22 Marzo 2013, ore 11:00-13:00 : The log minimal model program for M_g, II (Angelo Lopez)

Venerdi 5 Aprile 2013, ore 11:00-13:00 : The moduli space of pseudostable curves via GIT of 3-canonical curves (Fabio Felici)

Venerdi 12 Aprile 2013, ore 11:00-13:00 : The moduli space of pseudostable curves via GIT of 4-canonical curves (Fabio Felici);
The moduli space of pseudo-stable curves as the first step in the LMMP for M_g, I (Andrea Bruno).

Venerdi 19 Aprile 2013, ore 11:00-13:00 : The moduli space of pseudo-stable curves as the first step in the LMMP for M_g, II (Andrea Bruno).

Venerdi 3 Maggio 2013, ore 11:00-13:00 : The moduli space of h-semistable and c-semistable curves via Hilbert/Chow GIT of 2-canonical curves (Margarida Melo).

Venerdi 10 Maggio 2013, ore 11:00-13:00 : The first flip in the log minimal model program for M_g, I (Margarida Melo).

Venerdi 17 Maggio 2013, ore 11:00-13:00 : The first flip in the log minimal model program for M_g, II (Margarida Melo).

Venerdi 24 Maggio 2013, ore 11:00-13:00 : Local stable reduction for planar curve singularities (Andrea Bruno).

Giovedi 6 Giugno 2013, ore 15:00-17:00 : Heuristics for the critical values of the log minimal model program for M_g (Filippo Viviani).

PROGRAM

The log minimal model program for the moduli space of curves

ACADEMIC YEAR 2011/2012

TALKS

Giovedi 9 Febbraio 2012, ore 10:30-12:30 : The Picard group of $\ov{M}_{g,n}$ I (Filippo Viviani).

Giovedi 23 Febbraio 2012, ore 10:30-12:30 : The Picard group of $\ov{M}_{g,n}$ II (Filippo Viviani).

Giovedi 1 Marzo 2012, ore 14:00-16:00 : The Picard group of $\ov{M}_{g,n}$ III (Filippo Viviani).

Giovedi 8 Marzo 2012, ore 14:00-16:00 : The canonical class and Mumford relation (Edoardo Sernesi).

Giovedi 15 Marzo 2012, ore 14:00-16:00 : The slope of effective divisors. Explicit effective divisors and the Kodaira dimension (Edoardo Sernesi).

Giovedi 29 Marzo 2012, ore 10:00-12:00 : Ample and big cone of $\ov{M}_{g,n}$: results and conjectures of Gibney-Keel-Morrison I (Angelo Lopez).

Giovedi 12 Aprile 2012, ore 10:00-12:00 : Ample and big cone of $\ov{M}_{g,n}$: results and conjectures of Gibney-Keel-Morrison II (Angelo Lopez).

Giovedi 19 Aprile 2012, ore 14:00-16:00 : Ample and big cone of $\ov{M}_{g,n}$: results and conjectures of Gibney-Keel-Morrison III (Angelo Lopez).

Giovedi 3 Maggio 2012, ore 13:30-15:30 : The restricted nef cone and the weakly positive cone of $\ov{M}_g$: an overview of the results of Cornalba-Harris and Moriwaki (Filippo Viviani).

Giovedi 17 Maggio 2012, ore 11:00-13:00 : The weakly positive cone of $\ov{M}_g$: Moriwaki inequalities (Filippo Viviani).

Lunedi 21 Maggio 2012, ore 10:30-12:30 : The restricted nef cone of $\ov{M}_{g, n}$: Cornalba-Harris and Cornalba inequalities (Filippo Viviani).

Giovedi 31 Maggio 2012, ore 14:00-16:00 : Counterexamples related to the F-conjecture: Keel-Vermeire example and Pixton example (Andrea Bruno).

Giovedi 14 Giugno 2012, ore 14:00-16:00 : Lower bounds for the slope of effective divisors (Andrea Bruno).

PROGRAM

Cones of divisors of $\ov{M}_{g,n}$.

1. The Picard group of $\ov{M}_{g,n}$.

               References: [AC87], [AC09], [ACG, Chap. XIII, Chap. XIX], [Mor, Chap. 1].

2. The Pseudo-Effective cone via the slope.

       2.1 Lower Bounds via effective divisors: Brill-Noether divisors, Petri divisors, Koszul divisors.

                References: [HM82], [HM82], [EH87], [Far09c], [Far09a, Sec. 2.2, Sec. 3], [Far09b, Sec. 5], [Mor, Sec. 2.1, 2.2], [CFM].

       2.2 Upper bounds via moving curves.

                References: [HM98], [Pan], [Far09b, Sec. 4], [Mor, Sec. 2.3], [CFM].

2.3 Known results on the Kodaira dimension of $\ov{M}_{g,n}$.

                References: [Far09a], [Far09b], [Ver], [Log].

3. The Nef cone.

        3.1 The restricted Nef cone.

                References:[CH88], [ACG11, Chap. XIV], [Mor, Sec. 3.1].

   3.2 F-conjecture and the Nef dichotomy.

                References: [GKM02], [Mor, Sec. 3.2].

BIBLIOGRAPHY

BOOKS:

[ACG11] E. Arbarello, M. Cornalba, P. A. Griffiths: Geometry of algebraic curves. Volume II. With a contribution by Joseph Daniel Harris. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 268. Springer, Heidelberg, 2011.

[HM98] J. Harris, I. Morrison: Moduli of curves. Graduate text in mathematics 187. Springer-Verlag, New York-Heidelberg, 1998.

[Mor] I. Morrison: Mori theory of moduli spaces of stable curves. Book in preparation
(preliminary draft available at http://www.projectivepress.com/moduli/moristablecurves.pdf).

SURVEYS:

[AH] J. Alper, D. Hyeon: GIT construction of log canonical models of $\ov{M}_g$. Preprint available at arXiv:1109:2173.

[AH] J. Alper, D. Hyeon: GIT construction of log canonical models of $\ov{M}_g$. Preprint available at arXiv:1109:2173.

[CFM] D. Chen, G. Farkas, I. Morrison: Effective divisors on moduli spaces. Preprint available at arXiv:1205.6138.

[Far09a] G. Farkas: The global geometry of the moduli space of curves. Algebraic geometry—Seattle 2005. Part 1, 125–147, Proc. Sympos. Pure Math., 80, Part 1,
Amer. Math. Soc., Providence, RI, 2009 (available at arXiv:math/0612251).

[Far09b] G. Farkas: Birational aspects of the geometry of $M_g$. Surveys in differential geometry. Vol. XIV. Geometry of Riemann surfaces and their moduli spaces, 57–110,
Surv. Differ. Geom., 14, Int. Press, Somerville, MA, 2009 (available at arXiv:0810.0702).

[FS11] M. Fedorchuk, D. I. Smyth: Alternate compactifications of moduli space of curves. To appear in Handbook of Moduli (G. Farkas and I. Morrison, editors),
available at arXiv:1012.0329.

[Mor10] I. Morrison: GIT Constructions of Moduli Spaces of Stable Curves and Maps. Ji, Lizhen (ed.) et al., Geometry of Riemann surfaces and their moduli spaces.
Somerville, MA: International Press. Surveys in Differential Geometry 14 (2010), 315--369 (available at arXiv:0810.2340).

[Ver] A. Verra: Rational parametrizations of moduli spaces of curves. To appear in Handbook of Moduli (G. Farkas and I. Morrison, editors), available at arXiv:1112.6095.

PAPERS:

[AS] V. Alexeev, D. Swinarski: Nef divisors on $\ov{M}_{0,n}$ from GIT. Preprint available at arXiv:0812.0778.

[AFS] J. Alper, M. Fedorchuk, D. I. Smyth: Singularities with $\Gm$-action and the log minimal model program for $\ov{M}_g$. Preprint available at arXiv:1010.3751.

[ASvdW] J. Alper, D. I. Smyth, F. van der Wyck: Weakly proper moduli stacks of curves. Preprint available at arXiv:1012.0538.

[AC87] E. Arbarello, M. Cornalba: The Picard groups of the moduli spaces of curves. Topology 26 (1987), 153–-171.

[AC09] E. Arbarello, M. Cornalba: Divisors in the moduli spaces of curves. Surveys in differential geometry. Vol. XIV. Geometry of Riemann surfaces and their moduli spaces, 1–22,
Surv. Differ. Geom., 14, Int. Press, Somerville, MA, 2009 (available at arXiv:0810.5373).

[CH88] M. Cornalba, J. Harris: Divisor classes associated to families of stable varieties, with applications to the moduli space of curves. Ann. Sci. \'Ecole Norm. Sup. (4) 21 (1988), 455–475.

[EH87] D. Eisenbud, J. Harris: The Kodaira dimension of the moduli space of curves of genus $\geq 23$. Invent. Math. 90 (1987), 359–387.

[Far09c] G. Farkas: Koszul divisors on moduli spaces of curves. Amer. J. Math. 131 (2009), 819–867 (available at arXiv:math/0607475).

[Fed11] M. Fedorchuk: Moduli of weighted pointed stable curves and log canonical models of $\ov{M}_{g,n}$. Math. Res. Lett. 18 (2011), 663–675 (available at arXiv:1004.4938).

[Fed] M. Fedorchuk: Moduli spaces of hyperelliptic curves with A and D singularities. Preprint available at arXiv:1007.4828.

[FS11] M. Fedorchuk, D. I. Smyth: Ample divisors on moduli spaces of pointed rational curves. J. Algebraic Geom. 20 (2011), 599–629 (available at arXiv:0810.1677).

[GJM] N. Giansiracusa, D. Jensen, H.-B. Moon: GIT Compactifications of $M_{0,n}$ and Flips. Preprint available at arXiv:1112.0232.

[GKM02] A. Gibney, S. Keel, I. Morrison: Towards the ample cone of $\ov M_ {g,n}$. J. Amer. Math. Soc. 15 (2002), 273-294 (available at arXiv:math/0006208).

[HM90] J. Harris, I. Morrison: Slopes of effective divisors on the moduli space of stable curves. Invent. Math. 99 (1990), 321–355.

[HM82] J. Harris, D. Mumford: On the Kodaira dimension of the moduli space of curves. With an appendix by William Fulton. Invent. Math. 67 (1982), 23–88.

[Has03] B. Hassett: Moduli spaces of weighted pointed stable curves. Adv. Math. 173 (2003), 316–352 (available at arXiv:math/0205009).

[HH09] B. Hassett, D. Hyeon: Log canonical models for the moduli space of curves: first divisorial contraction. Trans. Amer. Math. Soc. 361 (2009), 4471--4489
(available at arXiv:math/0607477).

[HH] B. Hassett, D. Hyeon: Log canonical models for the moduli space of curves: the first flip. Preprint available at arXiv:0806.3444.

[Hye] D. Hyeon: An outline of the log minimal model program for the moduli space of curves. Preprint available at arXiv:1006.1094.

[HK00] Y. Hu, S. Keel: Mori dream spaces and GIT. Dedicated to William Fulton on the occasion of his 60th birthday. Michigan Math. J. 48 (2000), 331–348.

[HM10] D. Hyeon, I. Morrison: Stability of Tails and 4-Canonical Models. Math. Res. Lett. 17 (2010), 721--729 (available at arXiv:0806.1269).

[KM10] Y.-H. Kiem, H.-B. Moon: Moduli spaces of weighted pointed stable rational curves via GIT. Preprint available at arXiv:1002.2461.

[Log03] A. Logan: The Kodaira Dimension of Moduli Spaces of Curves with Marked Points. American Journal of Mathematics 125 (2003), 105--138.

[Moo1] H.-B. Moon: Log canonical models for the moduli space of pointed stable rational curves. Preprint available at arXiv:1101.1166.

[Moo2] H.-B. Moon: Log canonical models for $\ov{M}_{g,n}$. Preprint available at arXiv:1111.5354.

[Pan] R. Pandharipande: Descendent bounds for effective divisors on the moduli space of curves. Preprint available at arXiv:0805.0601.

[Sch91] D. Schubert: A new compactification of the moduli space of curves. Compositio Math. 78 (1991), 297--313.

[Smy] D. I. Smyth: Towards a classification of modular compactifications of the moduli space of curves (with an appendix of J. Hall). Preprint available at arXiv:0902.3690.

[Smy11] D. I. Smyth: Modular compactifications of the space of pointed elliptic curves I. Compositio Math. 147 (2011), 877--913 (available at arXiv:0808.0177).

[Smy12] D. I. Smyth: Modular compactifications of the space of pointed elliptic curves II. To appear in Compositio Math., available at arXiv:1005.1083.

[vdW10] F. van der Wyck: Moduli of singular curves and crimping. Ph.D. thesis, Harvard, 2010.