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## Hilbert schemes of lines and conics and automorphism groups of Fano threefolds

Cornell University
,
2016.

We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds with Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold of index 1 and genus 10. We also show that the action of the automorphism group of a Fano threefold X of index 2 (respectively, 1) on an irreducible component of its Hilbert scheme of lines (respectively, conics) is faithful if the anticanonical class of X is very ample with a possible exception of several explicit cases. We use these faithfulness results to prove finiteness of the automorphism groups of most Fano threefolds and classify explicitly all Fano threefolds with infinite automorphism group. We also discuss a derived category point of view on the Hilbert schemes of lines and conics, and use this approach to identify some of them.

Prokhorov Y., Kuznetsov A., Shramov K., Japanese Journal of Mathematics 2018 Vol. 13 No. 1 P. 109–185

We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds of Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold ...

Added: November 22, 2017

Cheltsov I., Przyjalkowski V., Shramov K., European Journal of Mathematics 2016 Vol. 2 No. 1 P. 96–119

We study quartic double solids admitting icosahedral symmetry. ...

Added: December 3, 2015

Galkin S., / Cornell University. Series math "arxiv.org". 2018. No. 1809.02705.

We classify smooth Fano threefolds that admit degenerations to toric Fano threefolds with ordinary double points. ...

Added: September 25, 2018

Buryak A., Feigin B. L., Nakajima H., International Mathematics Research Notices 2015 Vol. 2015 No. 13 P. 4708–4715

In a recent paper the first two authors proved that the generating series of the Poincare polynomials of the quasihomogeneous Hilbert schemes of points in the plane has a simple decomposition in an infinite product. In this paper we give a very short geometrical proof of that formula. ...

Added: September 29, 2020

Popov P., / Cornell University. Series math "arxiv.org". 2018. No. 1810.04563.

We study relations in the Grothendieck ring of varieties which connect the Hilbert scheme of points on a cubic hypersurface Y with a certain moduli space of twisted cubic curves on Y. These relations are generalizations of the "beautiful" Y-F(Y) relation by Galkin and Shinder which connects Y with the Hilbert scheme of two points on Y and the Fano variety F(Y) of lines on Y. We ...

Added: October 23, 2018

Galkin S., Popov P., / Cornell University. Series math "arxiv.org". 2018. No. 1810.07001.

Let X(n) denote n-th symmetric power of a cubic surface X. We show that X(4)×X is stably birational to X(3)×X, despite examples when X(4) is not stably birational to X(3). ...

Added: October 19, 2018

Galkin S., / Cornell University. Series math "arxiv.org". 2018. No. 1809.02738.

We show that G-Fano threefolds are mirror-modular.
1. Mirror maps are inversed reversed Hauptmoduln for moonshine subgroups of SL2(ℝ).
2. Quantum periods, shifted by an integer constant (eigenvalue of quantum operator on primitive cohomology) are expansions of weight 2 modular forms (theta-functions) in terms of inversed Hauptmoduln.
3. Products of inversed Hauptmoduln with some fractional powers of shifted ...

Added: September 25, 2018

Bogomolov F. A., Kulikov V. S., Central European Journal of Mathematics 2013 Vol. 11 No. 2 P. 254–263

The article contains a new proof that the Hilbert scheme of irreducible surfaces of degree m in ℙ m+1 is irreducible except m = 4. In the case m = 4 the Hilbert scheme consists of two irreducible components explicitly described in the article. The main idea of our approach is to use the proof ...

Added: November 21, 2012

Galkin S., Shinder E., / Cornell University. Series math "arxiv.org". 2014. No. 1405.5154.

We find a relation between a cubic hypersurface Y and its Fano variety of lines F(Y) in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational ...

Added: May 21, 2014

Vologodsky V., Finkelberg M. V., Bezrukavnikov R., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163–190

Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...

Added: December 17, 2015

Coates T., Corti A., Galkin S. et al., Geometry and Topology 2016 Vol. 20 No. 1 P. 103–256

The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. In this paper we compute the quantum periods of all 3-dimensional Fano manifolds. In particular we show that 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by ...

Added: November 18, 2014

Bezrukavnikov R., Finkelberg M. V., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163–190

Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...

Added: December 20, 2014

Kuznetsov A., Central European Journal of Mathematics 2012 Vol. 10 No. 4 P. 1198–1231

We introduce the notion of an instanton bundle on a Fano threefold of index 2. For such bundles we give an analogue of a monadic description and discuss the curve of jumping lines. The cases of threefolds of degree 5 and 4 are considered in a greater detail. ...

Added: September 10, 2012

Prokhorov Y., Sbornik Mathematics 2016 Vol. 207 No. 7 P. 983–1009

We study singular Fano threefolds of type $ V_{22}$. ...

Added: September 3, 2016

Gorsky E., Negut A., Journal de Mathématiques Pures and Appliquées 2015 Vol. 104 No. 3 P. 403–435

We consider the construction of refined Chern-Simons torus knot invariants by M. Aganagic and S. Shakirov from the DAHA viewpoint of I. Cherednik. We prove Cherednik's conjecture on the stabilization of superpolynomials, and then use the results of O. Schiffmann and E. Vasserot to relate knot invariants with the Hilbert scheme of points on the ...

Added: February 14, 2015

Gorsky E., Hogancamp M., / Cornell University. Series arXiv "math". 2017.

We define a deformation of the triply graded Khovanov-Rozansky homology of a link L depending on a choice of parameters for each component of L. We conjecture that this invariant restores the missing symmetry of the triply graded Khovanov-Rozansky homology, and in addition satisfies a number of predictions coming from a conjectural connection with Hilbert schemes of points ...

Added: December 28, 2017

Eugene Gorsky, Oblomkov A., Rasmussen J. et al., Duke Mathematical Journal 2014 Vol. 163 No. 14 P. 2709–2794

We conjecturally extract the triply graded Khovanov–Rozansky homology of the (m,n) torus knot from the unique finite-dimensional simple representation of the rational DAHA of type A, rank n-1, and central character m/n. The conjectural differentials of Gukov, Dunfield, and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the ...

Added: December 9, 2014

Gorsky E., Negut A., Rasmussen J., / Cornell University. Series arXiv "math". 2016.

We construct a categorification of the maximal commutative subalgebra of the type A Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functor allows one to match the Hochschild homology of ...

Added: September 19, 2016

Gorsky Evgeny, Mazin M., Journal of Combinatorial Theory, Series A 2013 Vol. 120 No. 1 P. 49–63

J. Piontkowski described the homology of the Jacobi factor of a plane curve singularity with one Puiseux pair. We discuss the combinatorial structure of his answer, in particular, relate it to the bigraded deformation of Catalan numbers introduced by A. Garsia and M. Haiman. ...

Added: December 9, 2014

Prokhorov Y., Kuznetsov A., / Cornell University. Series arXiv "math". 2017.

We give an explicit construction of prime Fano threefolds of genus 12 with a G_m-action, describe their isomorphism classes and automorphism groups. ...

Added: November 29, 2017

Buryak A., Feigin B. L., , in: <i>Symmetries, Integrable Systems and Representations</i>Vol. 40: Symmetries, Integrable Systems and Representations. Springer, 2013.

In this paper we prove that the generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes of points in the plane has a beautiful decomposition into an infinite product. We also compute the generating series of the numbers of quasihomogeneous components in a moduli space of sheaves on the projective plane. The answer is ...

Added: September 30, 2020

Gorsky E., Geometry and Topology 2018 Vol. 22 P. 645–691

We conjecture an expression for the dimensions of the Khovanov–Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture (2012) relating the HOMFLY polynomial to the Euler numbers of the ...

Added: August 21, 2018

Cheltsov I., Shramov K., / Cornell University. Series math "arxiv.org". 2018.

We study the problem of existence of K\"ahler--Einstein metrics on smooth Fano threefolds of Picard rank one and anticanonical degree 22 that admit a faithful action of the multiplicative group ℂ∗. We prove that, except possibly two explicitly described cases, all such smooth Fano threefolds are K\"ahler--Einstein. ...

Added: October 21, 2018

Cheltsov I., Park J., Ahmadinezhad H., / Cornell University. Series arXiv "math". 2020.

We prove that a quasi-smooth Fano threefold hypersurface is birationally rigid if and only if it has Fano index one. ...

Added: August 19, 2020