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.