On universal Severi varieties of low genus K3 surfaces

Ciliberto, Ciro; Dedieu, Thomas

doi:10.1007/s00209-011-0898-3

Cited by 17 publications

(39 citation statements)

References 31 publications

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“…The stack

V_{g, δ, l}

is smooth and every irreducible component has dimension

19 + l + (g - δ)

. It was conjectured by Ciliberto and Dedieu [, Theorem 2.1] that the quotient

{scriptV}_{g, δ} / {normalΣ}_{δ}

is always irreducible and they proved it in the range

3 ⩽ g ⩽ 11

g \neq 10

and

0 ⩽ δ ⩽ g

. The forgetful map

π_{δ} : {scriptV}_{g, δ} / {normalΣ}_{δ} \to {scriptF}_{g}

is smooth and dominant when restricted to any irreducible component; see [, Proposition 4.8].…”

Section: Nodal Curves On K3 Surfacesmentioning

confidence: 99%

“…Note that for a scheme

S

and

S

‐schemes

X, Y, Z

, if

X \to Y

is a birational

S

‐map, the same holds for the induced map

X \times_{S} Z \to Y \times_{S} Z .

Therefore, it is enough to prove that the general fiber of

c_{11, δ} : {scriptV}_{11, δ} \to {scriptM}_{11 - δ, [2 δ]}

is irreducible. It is shown in the proof of [, Theorem 2.1] that the general fiber of

{scriptV}_{g, δ} / {normalΣ}_{δ} \to {scriptM}_{g - δ}

is irreducible for

δ

and

g

in certain range. This result can be easily extended to the fibers of

c_{g, δ}

.…”

Section: Parameter Spaces Of Flags and Degeneration To Projective Conesmentioning

confidence: 99%

“…An open subset of

V_{g}

is a

{double-struckP}^{g}

‐bundle over

F_{g}

. Note that

dim ({Hilb}_{g}) = 4 g - 4 + g^{2} and dim (V_{g}) = 18 + g + {(g + 1)}^{2} .

The natural forgetful maps sit in the following diagram:

Our situation is slightly different from , since we need to keep track of the nodes. For

0 ⩽ δ ⩽ g - 2

, we define the incidence variety

{boldV}_{g, δ} \subset {boldV}_{g} \times {(P^{g})}^{δ}

to be the closure of the set of points

(S, X, x_{1}, \dots, x_{δ})

with

X

irreducible

δ

‐nodal with nodes at

x_{1}, \dots, x_{δ}

.…”

Section: Parameter Spaces Of Flags and Degeneration To Projective Conesmentioning

confidence: 99%

“…The first was proved by and states that a smooth

K 3

surface

S \subset P^{g}

can flatly degenerate inside

{double-struckP}^{g}

to a projective cone over any hyperplane section

S_{X} \subset P^{g}

. Moreover (see [, Lemma 2.3]), one can do this inside the fiber of

p_{g, δ}

. Thus, if

S_{X}

is the projective cone over the nodal curve

X \subset S

inside

{double-struckP}^{g}

, then

(S_{X}, X, x_{1}, \dots, x_{δ})

lies in every irreducible component of the general fiber of

p_{g, δ}

.…”

Section: Parameter Spaces Of Flags and Degeneration To Projective Conesmentioning

confidence: 99%

“…The map

\begin{matrix} right {boldV}_{g, δ} & left \to & left V_{g, δ} / Σ_{δ} \\ right false(S, X, x_{1}, \dots, x_{δ} false) & left \mapsto & left false(S, X false) \end{matrix}

is étale and the tangent space of the fiber at

(S_{X}, X)

(see [, Lemma 2.4; , § 4.5.2]) is isomorphic to

H^{0} false(S_{X}, N_{S_{X} / {double-struckP}^{g}} (- 1) false) ≅ ⨁_{k ⩾ 1} H^{0} false(X, N_{X / {double-struckP}^{g}} (- k) false) .

The computation of these cohomology groups was done in [, § 3] by specializing to a canonical graph curve , namely, the union of

2 g - 2

lines in

{double-struckP}^{g - 1}

, each meeting three others at distinct points. The dual graph of such a curve is a trivalent graph with

2 g - 2

nodes and

3 g - 3

edges.…”

Section: Parameter Spaces Of Flags and Degeneration To Projective Conesmentioning

confidence: 99%

See 4 more Smart Citations

Geometry of the moduli space of n-pointed K3 surfaces of genus 11

Barros

2018

Bull. London Math. Soc.

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“…The stack

V_{g, δ, l}

is smooth and every irreducible component has dimension

19 + l + (g - δ)

. It was conjectured by Ciliberto and Dedieu [, Theorem 2.1] that the quotient

{scriptV}_{g, δ} / {normalΣ}_{δ}

is always irreducible and they proved it in the range

3 ⩽ g ⩽ 11

g \neq 10

and

0 ⩽ δ ⩽ g

. The forgetful map

π_{δ} : {scriptV}_{g, δ} / {normalΣ}_{δ} \to {scriptF}_{g}

is smooth and dominant when restricted to any irreducible component; see [, Proposition 4.8].…”

Section: Nodal Curves On K3 Surfacesmentioning

confidence: 99%

“…Note that for a scheme

S

and

S

‐schemes

X, Y, Z

, if

X \to Y

is a birational

S

‐map, the same holds for the induced map

X \times_{S} Z \to Y \times_{S} Z .

Therefore, it is enough to prove that the general fiber of

c_{11, δ} : {scriptV}_{11, δ} \to {scriptM}_{11 - δ, [2 δ]}

is irreducible. It is shown in the proof of [, Theorem 2.1] that the general fiber of

{scriptV}_{g, δ} / {normalΣ}_{δ} \to {scriptM}_{g - δ}

is irreducible for

δ

and

g

in certain range. This result can be easily extended to the fibers of

c_{g, δ}

.…”

Section: Parameter Spaces Of Flags and Degeneration To Projective Conesmentioning

confidence: 99%

“…An open subset of

V_{g}

is a

{double-struckP}^{g}

‐bundle over

F_{g}

. Note that

dim ({Hilb}_{g}) = 4 g - 4 + g^{2} and dim (V_{g}) = 18 + g + {(g + 1)}^{2} .

The natural forgetful maps sit in the following diagram:

Our situation is slightly different from , since we need to keep track of the nodes. For

0 ⩽ δ ⩽ g - 2

, we define the incidence variety

{boldV}_{g, δ} \subset {boldV}_{g} \times {(P^{g})}^{δ}

to be the closure of the set of points

(S, X, x_{1}, \dots, x_{δ})

with

X

irreducible

δ

‐nodal with nodes at

x_{1}, \dots, x_{δ}

.…”

Section: Parameter Spaces Of Flags and Degeneration To Projective Conesmentioning

confidence: 99%

“…The first was proved by and states that a smooth

K 3

surface

S \subset P^{g}

can flatly degenerate inside

{double-struckP}^{g}

to a projective cone over any hyperplane section

S_{X} \subset P^{g}

. Moreover (see [, Lemma 2.3]), one can do this inside the fiber of

p_{g, δ}

. Thus, if

S_{X}

is the projective cone over the nodal curve

X \subset S

inside

{double-struckP}^{g}

, then

(S_{X}, X, x_{1}, \dots, x_{δ})

lies in every irreducible component of the general fiber of

p_{g, δ}

.…”

Section: Parameter Spaces Of Flags and Degeneration To Projective Conesmentioning

confidence: 99%

“…The map

\begin{matrix} right {boldV}_{g, δ} & left \to & left V_{g, δ} / Σ_{δ} \\ right false(S, X, x_{1}, \dots, x_{δ} false) & left \mapsto & left false(S, X false) \end{matrix}

is étale and the tangent space of the fiber at

(S_{X}, X)

(see [, Lemma 2.4; , § 4.5.2]) is isomorphic to

H^{0} false(S_{X}, N_{S_{X} / {double-struckP}^{g}} (- 1) false) ≅ ⨁_{k ⩾ 1} H^{0} false(X, N_{X / {double-struckP}^{g}} (- k) false) .

The computation of these cohomology groups was done in [, § 3] by specializing to a canonical graph curve , namely, the union of

2 g - 2

lines in

{double-struckP}^{g - 1}

, each meeting three others at distinct points. The dual graph of such a curve is a trivalent graph with

2 g - 2

nodes and

3 g - 3

edges.…”

Section: Parameter Spaces Of Flags and Degeneration To Projective Conesmentioning

confidence: 99%

See 3 more Smart Citations

Geometry of the moduli space of n-pointed K3 surfaces of genus 11

Barros

2018

Bull. London Math. Soc.

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Limits of Pluri–Tangent Planes to Quartic Surfaces

Ciliberto

Dedieu

2014

Algebraic and Complex Geometry

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We describe, for various degenerations S → ∆ of quartic K3 surfaces over the complex unit disk (e.g., to the union of four general planes, and to a general Kummer surface), the limits as t ∈ ∆ * tends to 0 of the Severi varieties V δ (St), parametrizing irreducible δ-nodal plane sections of St. We give applications of this to (i) the counting of plane nodal curves through base points in special position, (ii) the irreducibility of Severi varieties of a general quartic surface, and (iii) the monodromy of the universal family of rational curves on quartic K3 surfaces.

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