## Final Report Summary - SMOOTH (Smoothness of the invariant Hilbert scheme of affine spherical varieties for the existence of wonderful varieties)

Spherical varieties are special complex algebraic varieties with an action of a linear algebraic group. More precisely, they are normal algebraic varieties with an action of a connected reductive algebraic group with an open (dense) orbit of a Borel subgroup.

Spherical varieties form a wide class among notable algebraic varieties arising in nature, in particular, the symmetric varieties (the algebraic analogue of Riemannian symmetric spaces) are spherical, and their theory has been developed in the last 25 years in many aspects; see for example:

- M. Brion, D. Luna, T. Vust, Espaces homogènes sphériques, Invent. Math., 1986;

- M. Brion, D. Luna, Sur la structure locale des variétés sphériques, Bull. Soc. Math. France, 1987;

- M. Brion, Groupe de Picard et nombre caractéristiques des variétés sphériques, Duke Math. J., 1989;

- F. Knop, On the set of orbits for a Borel subgroup, Comment. Math. Helv., 1995;

- D. Luna, Variétés sphériques de type A, Publ. Math. Inst. Hautes Études Sci., 2001.

Our main aim was the classification of spherical varieties: an open problem. The problem can be reduced to the classification of a special class of spherical varieties, called wonderful, and there was a conjecture (known as Luna's conjecture) for a complete solution of the problem [D. Luna, loc. cit., 2001]. Luna's conjecture has been partially proved under special hypotheses, see:

- D. Luna, loc. cit., 2001;

- G. Pezzini, Wonderful varieties of type C, PhD thesis, Università La Sapienza Roma, 2004;

- P.B. G. Pezzini, Wonderful varieties of type D, Represent. Theory, 2005;

- P.B. Wonderful varieties of type E, Represent. Theory, 2007;

- P.B. S. Cupit-Foutou, Equivariant deformations of the affine multicone over a flag variety, Adv. Math., 2008;

- I.V. Losev, Uniqueness property for spherical homogeneous spaces, Duke. Math. J., 2009;

- P.B. S. Cupit-Foutou, Classification of strict wonderful varieties, to appear on Ann. Inst. Fourier (Grenoble).

We planned to use the approach to the classification problem via invariant Hilbert schemes, introduced by V. Alexeev and M. Brion [J. Alg. Geom., 2005], generalising [P.B. S. Cupit-Foutou, loc. cit., 2008]. But one year later, at the starting date of our work, that method of research was already in use by S. Cupit-Foutou at an advanced stage of development; see:

- S. Cupit-Foutou, Invariant Hilbert schemes and wonderful varieties, arXiv:0811.1567v2 , 2009;

- S. Cupit-Foutou, Wonderful varieties: a geometrical realisation, arXiv:0907.2852v1 , 2009.

We then decided to keep the same objective (the classification of spherical varieties via Luna's conjecture) but using different methods.

We have extensively analysed the combinatorics of the involved objects, the so-called spherical systems. This has allowed us to better understand the interplay of combinatorics and geometry of wonderful varieties, and to start developing a more complete theory of wonderful varieties; see:

- P.B. D. Luna, An introduction to wonderful varieties with many examples of type F4, arXiv:0812.2340v2 , 2009.

We have solved some technical problems arising in generalising Luna's original approach to the classification and fixed the strategy for a full proof of Luna's conjecture, see:

- P.B. Primitive spherical systems, arXiv:0909.3765v1 , 2009.

- P.B. G. Pezzini, Wonderful varieties of type B and C, arXiv:0909.3771v1 , 2009.

The work performed during this project has led to a constructive approach to the classification, which essentially provide an algorithm to associate a wonderful subgroup (i.e. the generic stabiliser of a wonderful variety) to a given spherical system. We have some partial results, but the theoretical investigation is still going on. We have made this public by:

- wonderful varieties and spherical orbits in simple projective spaces, talk at 'Invariant Hilbert schemes and wonderful varieties', workshop at Mathematisches Institut Universitaet Basel, 2009.

Spherical varieties form a wide class among notable algebraic varieties arising in nature, in particular, the symmetric varieties (the algebraic analogue of Riemannian symmetric spaces) are spherical, and their theory has been developed in the last 25 years in many aspects; see for example:

- M. Brion, D. Luna, T. Vust, Espaces homogènes sphériques, Invent. Math., 1986;

- M. Brion, D. Luna, Sur la structure locale des variétés sphériques, Bull. Soc. Math. France, 1987;

- M. Brion, Groupe de Picard et nombre caractéristiques des variétés sphériques, Duke Math. J., 1989;

- F. Knop, On the set of orbits for a Borel subgroup, Comment. Math. Helv., 1995;

- D. Luna, Variétés sphériques de type A, Publ. Math. Inst. Hautes Études Sci., 2001.

Our main aim was the classification of spherical varieties: an open problem. The problem can be reduced to the classification of a special class of spherical varieties, called wonderful, and there was a conjecture (known as Luna's conjecture) for a complete solution of the problem [D. Luna, loc. cit., 2001]. Luna's conjecture has been partially proved under special hypotheses, see:

- D. Luna, loc. cit., 2001;

- G. Pezzini, Wonderful varieties of type C, PhD thesis, Università La Sapienza Roma, 2004;

- P.B. G. Pezzini, Wonderful varieties of type D, Represent. Theory, 2005;

- P.B. Wonderful varieties of type E, Represent. Theory, 2007;

- P.B. S. Cupit-Foutou, Equivariant deformations of the affine multicone over a flag variety, Adv. Math., 2008;

- I.V. Losev, Uniqueness property for spherical homogeneous spaces, Duke. Math. J., 2009;

- P.B. S. Cupit-Foutou, Classification of strict wonderful varieties, to appear on Ann. Inst. Fourier (Grenoble).

We planned to use the approach to the classification problem via invariant Hilbert schemes, introduced by V. Alexeev and M. Brion [J. Alg. Geom., 2005], generalising [P.B. S. Cupit-Foutou, loc. cit., 2008]. But one year later, at the starting date of our work, that method of research was already in use by S. Cupit-Foutou at an advanced stage of development; see:

- S. Cupit-Foutou, Invariant Hilbert schemes and wonderful varieties, arXiv:0811.1567v2 , 2009;

- S. Cupit-Foutou, Wonderful varieties: a geometrical realisation, arXiv:0907.2852v1 , 2009.

We then decided to keep the same objective (the classification of spherical varieties via Luna's conjecture) but using different methods.

We have extensively analysed the combinatorics of the involved objects, the so-called spherical systems. This has allowed us to better understand the interplay of combinatorics and geometry of wonderful varieties, and to start developing a more complete theory of wonderful varieties; see:

- P.B. D. Luna, An introduction to wonderful varieties with many examples of type F4, arXiv:0812.2340v2 , 2009.

We have solved some technical problems arising in generalising Luna's original approach to the classification and fixed the strategy for a full proof of Luna's conjecture, see:

- P.B. Primitive spherical systems, arXiv:0909.3765v1 , 2009.

- P.B. G. Pezzini, Wonderful varieties of type B and C, arXiv:0909.3771v1 , 2009.

The work performed during this project has led to a constructive approach to the classification, which essentially provide an algorithm to associate a wonderful subgroup (i.e. the generic stabiliser of a wonderful variety) to a given spherical system. We have some partial results, but the theoretical investigation is still going on. We have made this public by:

- wonderful varieties and spherical orbits in simple projective spaces, talk at 'Invariant Hilbert schemes and wonderful varieties', workshop at Mathematisches Institut Universitaet Basel, 2009.