Extremal Rays and Automorphisms of Holomorphic Symplectic Varieties

Hassett, Brendan; Tschinkel, Yuri

doi:10.1007/978-3-319-29959-4_4

Cited by 18 publications

(23 citation statements)

References 20 publications

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“…After a change of base to

(h_{1}, h_{2}, θ)

we obtain

[\begin{matrix} 0 & 2 & 0 \\ 2 & 0 & 0 \\ 0 & 0 & - 10 \end{matrix}] .

We find that the map

ξ

is given by

h_{1} + h_{2}

. Since there is a divisor with self‐intersection

- 10

and divisibility 2 perpendicular to

h_{1} + h_{2}

, it follows that

ξ

contracts a

P^{2}

to a point (see [, § 5.1] or [, § 2]). We can identify this

P^{2}

as the set of pairs of points on

S_{A}

such that the line spanned by these points is contained in the threefold section of

G (2, 5)

containing

S_{A}

.…”

Section: First Construction — Singular Epw Cubesmentioning

confidence: 98%

Hyper-Kähler fourfolds and Kummer surfaces

Iliev

Kapustka

et al. 2017

Proc. London Math. Soc.

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Abstract. We show that a Hilbert scheme of conics on a Fano fourfold double cover of P 2 × P 2 ramified along a divisor of bidegree (2, 2) admits a P 1 -fibration with base being a hyper-Kähler fourfold. We investigate the geometry of such fourfolds relating them with degenerated EPW cubes, with elements in the Brauer groups of K3 surfaces of degree 2, and with Verra threefolds studied in [Ver04]. These hyper-Kähler fourfolds admit natural involutions and complete the classification of geometric realizations of anti-symplectic involutions on hyper-Kähler 4-folds of type K3 [2] . As a consequence we present also three constructions of quartic Kummer surfaces in P 3 : as Lagrangian and symmetric degeneracy loci and as the base of a fibration of conics in certain threefold quadric bundles over P 1 .

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“…After a change of base to

(h_{1}, h_{2}, θ)

we obtain

[\begin{matrix} 0 & 2 & 0 \\ 2 & 0 & 0 \\ 0 & 0 & - 10 \end{matrix}] .

We find that the map

ξ

is given by

h_{1} + h_{2}

. Since there is a divisor with self‐intersection

- 10

and divisibility 2 perpendicular to

h_{1} + h_{2}

, it follows that

ξ

contracts a

P^{2}

to a point (see [, § 5.1] or [, § 2]). We can identify this

P^{2}

as the set of pairs of points on

S_{A}

such that the line spanned by these points is contained in the threefold section of

G (2, 5)

containing

S_{A}

.…”

Section: First Construction — Singular Epw Cubesmentioning

confidence: 98%

Hyper-Kähler fourfolds and Kummer surfaces

Iliev

Kapustka

et al. 2017

Proc. London Math. Soc.

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“…For

n = 3

, in [, Example 14] Hassett and Tschinkel proved the existence of a non‐natural automorphism

f \in Aut (S^{[3]})

with

T_{f} ≅ false⟨ 2 false⟩

when S is a K 3 surface with

Pic (S) = Z H

H^{2} = 114

(i.e.

t = 57

).…”

Section: Numerical Conditionsmentioning

confidence: 99%

“…In the case where is a non-trivial product of two integers and = 1, we have the following result. (for more details, see [12,Chapter XXXIII,§ 18]). We define the fundamental solution in an equivalence class to be the solution with smallest nonnegative , if it is unique.…”

Section: Pell's Equationsmentioning

confidence: 99%

“…The case = 2, = 2 is the only one where a geometric construction of the involution is known: it is Beauville's involution on the Hilbert square of a quartic surface in ℙ 3 which does not contain any lines (see [3, § 6]). For = 3, in [18,Example 14] Hassett and Tschinkel proved the existence of a non-natural automorphism ∈ Aut ( [3] ) with ≅ ⟨2⟩ when is a 3 surface with Pic( ) = ℤ , 2 = 114 (i.e. = 57).…”

Section: Numerical Conditionsmentioning

confidence: 99%

See 1 more Smart Citation

Automorphisms of Hilbert schemes of points on a generic projective K3 surface

Cattaneo

2019

Mathematische Nachrichten

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We study automorphisms of the Hilbert scheme of n points on a generic projective K3 surface S, for any n≥2. We show that Aut(S[n]) is either trivial or generated by a non‐symplectic involution and we determine numerical and divisorial conditions which allow us to distinguish between the two cases. As an application of these results we prove that, for any n≥2, there exist infinitely many admissible degrees for the polarization of the K3 surface S such that Sfalse[nfalse] admits a non‐natural involution. This provides a generalization of the results of [7] for n=2.

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“…In any given example, one can also attempt to study the birational geometry of the contraction in order to obtain a result analogous to Theorem 8.2; this has been done systematically up to dimension 10 in [HT15] (along with other applications).…”

Section: Birational Geometry Of Moduli Spaces Of Sheaves: a Quick Surveymentioning

confidence: 99%

Wall-crossing implies Brill-Noether: Applications of stability conditions on surfaces

Bayer¹

2018

Algebraic Geometry: Salt Lake City 2015

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Over the last few years, wall-crossing for Bridgeland stability conditions has led to a large number of results in algebraic geometry, particular on birational geometry of moduli spaces.We illustrate some of the methods behind these result by reproving Lazarsfeld's Brill-Noether theorem for curves on K3 surfaces via wall-crossing. We conclude with a survey of recent applications of stability conditions on surfaces.The intended reader is an algebraic geometer with a limited working knowledge of derived categories.

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Extremal Rays and Automorphisms of Holomorphic Symplectic Varieties

Cited by 18 publications

References 20 publications

Hyper-Kähler fourfolds and Kummer surfaces

Hyper-Kähler fourfolds and Kummer surfaces

Automorphisms of Hilbert schemes of points on a generic projective K3 surface

Wall-crossing implies Brill-Noether: Applications of stability conditions on surfaces

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