Dual graphs of projective schemes

Dual graphs of projective schemes

Matteo Varbaro (University of Genova)

29/09/2015, Osnabr¨ uck, Germany

Example: 27 lines

Let us take 27 lines L

, . . . , L

in P

, and consider their union

C =

27

[

i =1

L

⊆ P

.

Let I

= {f ∈ C[x

, x

, x

, x

] : f (P) = 0 ∀ P ∈ C }.

Which constrains must we have on the configuration of the lines in order that I

⊆ C[x

, x

, x

, x

] is generated by 2 polynomials (that is C ⊆ P

is a complete intersection)?

As a consequence of the results I will present today, in this case it must happen that each line meets at least 10 of the others...

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Example: 27 lines

Let us take 27 lines L

, . . . , L

in P

, and consider their union

C =

27

[

i =1

L

⊆ P

.

Let I

= {f ∈ C[x

, x

, x

, x

] : f (P) = 0 ∀ P ∈ C }.

Which constrains must we have on the configuration of the lines in order that I

⊆ C[x

, x

, x

, x

] is generated by 2 polynomials (that is C ⊆ P

is a complete intersection)?

As a consequence of the results I will present today, in this case it

must happen that each line meets at least 10 of the others...

Example: 27 lines

Let us take 27 lines L

, . . . , L

in P

, and consider their union

C =

27

[

i =1

L

⊆ P

.

Let I

= {f ∈ C[x

, x

, x

, x

] : f (P) = 0 ∀ P ∈ C }.

Which constrains must we have on the configuration of the lines in order that I

⊆ C[x

, x

, x

, x

] is generated by 2 polynomials (that is C ⊆ P

is a complete intersection)?

As a consequence of the results I will present today, in this case it must happen that each line meets at least 10 of the others...

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Example: 27 lines

Let us take 27 lines L

, . . . , L

in P

, and consider their union

C =

27

[

i =1

L

⊆ P

.

Let I

= {f ∈ C[x

, x

, x

, x

] : f (P) = 0 ∀ P ∈ C }.

Which constrains must we have on the configuration of the lines in order that I

⊆ C[x

, x

, x

, x

] is generated by 2 polynomials (that is C ⊆ P

is a complete intersection)?

As a consequence of the results I will present today, in this case it

must happen that each line meets at least 10 of the others...

Example: 27 lines

On the other hand, as we know, on a smooth cubic X ⊆ P

there are exactly 27 lines, which can be read from the fact that X is the blow-up of P

along 6 generic points.

If C ⊆ P

is the union of such 27 lines, it is easy to see that I

= (f , g ), where f is the cubic defining X and g is a product of 9 linear forms. So C ⊆ P

is a complete intersection.

From the “blow-up interpretation”, it is immediate to check that the line corresponding to the exceptional divisor of any of the 6 points meet exactly 10 of the others.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Example: 27 lines

On the other hand, as we know, on a smooth cubic X ⊆ P

there are exactly 27 lines, which can be read from the fact that X is the blow-up of P

along 6 generic points.

If C ⊆ P

is the union of such 27 lines, it is easy to see that I

= (f , g ), where f is the cubic defining X and g is a product of 9 linear forms. So C ⊆ P

is a complete intersection.

From the “blow-up interpretation”, it is immediate to check that

the line corresponding to the exceptional divisor of any of the 6

points meet exactly 10 of the others.

Example: 27 lines

On the other hand, as we know, on a smooth cubic X ⊆ P

there are exactly 27 lines, which can be read from the fact that X is the blow-up of P

along 6 generic points.

If C ⊆ P

is the union of such 27 lines, it is easy to see that I

= (f , g ), where f is the cubic defining X and g is a product of 9 linear forms. So C ⊆ P

is a complete intersection.

From the “blow-up interpretation”, it is immediate to check that the line corresponding to the exceptional divisor of any of the 6 points meet exactly 10 of the others.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Motivations

Let X ⊆ P

be a projective scheme over K = K .

The main motivation for this talk comes from the desire of understanding how global properties of X ⊆ P

influence the combinatorial configuration of its irreducible components. One way to make precise the concept of “combinatorial

configuration of its irreducible components” is by meaning of the

dual graph of X ....

Motivations

Let X ⊆ P

be a projective scheme over K = K .

The main motivation for this talk comes from the desire of understanding how global properties of X ⊆ P

influence the combinatorial configuration of its irreducible components.

One way to make precise the concept of “combinatorial

configuration of its irreducible components” is by meaning of the dual graph of X ....

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Motivations

Let X ⊆ P

be a projective scheme over K = K .

The main motivation for this talk comes from the desire of understanding how global properties of X ⊆ P

influence the combinatorial configuration of its irreducible components.

One way to make precise the concept of “combinatorial

configuration of its irreducible components” is by meaning of the

dual graph of X ....

Definition of dual graph

Given X ⊆ P

, if X

, . . . , X

are its irreducible components, we form the dual graph G (X ) as follows:

The vertex set of G (X ) is {1, . . . , s}.

Two vertices i 6= j are connected by an edge if and only if: dim(X

∩ X

) = dim(X ) − 1.

From now on we will consider only equidimensional schemes. NOTE: If X is a projective curve, then {i , j } is an edge if and only if X

∩ X

6= ∅ (the empty set has dimension −1). If dim(X ) > 1, by intersecting X ⊆ P

with a generic hyperplane, we get a projective scheme in P

of dimension one less, and same dual graph! Iterating this trick we can often reduce questions to curves.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Definition of dual graph

Given X ⊆ P

, if X

, . . . , X

are its irreducible components, we form the dual graph G (X ) as follows:

The vertex set of G (X ) is {1, . . . , s}.

Two vertices i 6= j are connected by an edge if and only if: dim(X

∩ X

) = dim(X ) − 1.

From now on we will consider only equidimensional schemes.

NOTE: If X is a projective curve, then {i , j } is an edge if and only

if X

∩ X

6= ∅ (the empty set has dimension −1). If dim(X ) > 1,

by intersecting X ⊆ P

with a generic hyperplane, we get a

projective scheme in P

of dimension one less, and same dual

graph! Iterating this trick we can often reduce questions to curves.

Definition of dual graph

Given X ⊆ P

, if X

, . . . , X

are its irreducible components, we form the dual graph G (X ) as follows:

The vertex set of G (X ) is {1, . . . , s}.

Two vertices i 6= j are connected by an edge if and only if:

dim(X

∩ X

) = dim(X ) − 1.

From now on we will consider only equidimensional schemes. NOTE: If X is a projective curve, then {i , j } is an edge if and only if X

∩ X

6= ∅ (the empty set has dimension −1). If dim(X ) > 1, by intersecting X ⊆ P

with a generic hyperplane, we get a projective scheme in P

of dimension one less, and same dual graph! Iterating this trick we can often reduce questions to curves.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Definition of dual graph

Given X ⊆ P

, if X

, . . . , X

are its irreducible components, we form the dual graph G (X ) as follows:

The vertex set of G (X ) is {1, . . . , s}.

Two vertices i 6= j are connected by an edge if and only if:

dim(X

∩ X

) = dim(X ) − 1.

From now on we will consider only equidimensional schemes.

NOTE: If X is a projective curve, then {i , j } is an edge if and only

if X

∩ X

6= ∅ (the empty set has dimension −1). If dim(X ) > 1,

by intersecting X ⊆ P

with a generic hyperplane, we get a

projective scheme in P

of dimension one less, and same dual

graph! Iterating this trick we can often reduce questions to curves.

Definition of dual graph

Given X ⊆ P

, if X

, . . . , X

are its irreducible components, we form the dual graph G (X ) as follows:

The vertex set of G (X ) is {1, . . . , s}.

Two vertices i 6= j are connected by an edge if and only if:

dim(X

∩ X

) = dim(X ) − 1.

From now on we will consider only equidimensional schemes.

NOTE: If X is a projective curve, then {i , j } is an edge if and only if X

∩ X

6= ∅ (the empty set has dimension −1).

If dim(X ) > 1, by intersecting X ⊆ P

with a generic hyperplane, we get a projective scheme in P

of dimension one less, and same dual graph! Iterating this trick we can often reduce questions to curves.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Definition of dual graph

Given X ⊆ P

, if X

, . . . , X

are its irreducible components, we form the dual graph G (X ) as follows:

The vertex set of G (X ) is {1, . . . , s}.

Two vertices i 6= j are connected by an edge if and only if:

dim(X

∩ X

) = dim(X ) − 1.

From now on we will consider only equidimensional schemes.

NOTE: If X is a projective curve, then {i , j } is an edge if and only if X

∩ X

6= ∅ (the empty set has dimension −1). If dim(X ) > 1, by intersecting X ⊆ P

with a generic hyperplane, we get a projective scheme in P

of dimension one less, and same dual graph!

Iterating this trick we can often reduce questions to curves.

Definition of dual graph

Given X ⊆ P

, if X

, . . . , X

are its irreducible components, we form the dual graph G (X ) as follows:

The vertex set of G (X ) is {1, . . . , s}.

Two vertices i 6= j are connected by an edge if and only if:

dim(X

∩ X

) = dim(X ) − 1.

From now on we will consider only equidimensional schemes.

NOTE: If X is a projective curve, then {i , j } is an edge if and only if X

∩ X

6= ∅ (the empty set has dimension −1). If dim(X ) > 1, by intersecting X ⊆ P

with a generic hyperplane, we get a projective scheme in P

of dimension one less, and same dual graph! Iterating this trick we can often reduce questions to curves.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Hartshorne’s connectedness theorem

Given X ⊆ P

and the unique saturated homogeneous ideal I

⊆ S = K [x

, . . . , x

] s.t. X = Proj(S /I

), let us recall that X ⊆ P

is arithmetically Cohen-Macaulay (resp. arithmetically Gorenstein) if S /I

is Cohen–Macaulay (resp. Gorenstein).

A classical result by Hartshorne is that aCM schemes are connected in codimension one:

Hartshorne’s connectedness theorem

If X ⊆ P

is aCM, then G (X ) is a connected graph.

On the other hand ...

Hartshorne’s connectedness theorem

Given X ⊆ P

and the unique saturated homogeneous ideal I

⊆ S = K [x

, . . . , x

] s.t. X = Proj(S /I

), let us recall that X ⊆ P

is arithmetically Cohen-Macaulay (resp. arithmetically Gorenstein) if S /I

is Cohen–Macaulay (resp. Gorenstein).

A classical result by Hartshorne is that aCM schemes are connected in codimension one:

Hartshorne’s connectedness theorem

If X ⊆ P

is aCM, then G (X ) is a connected graph.

On the other hand ...

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Hartshorne’s connectedness theorem

Given X ⊆ P

and the unique saturated homogeneous ideal I

⊆ S = K [x

, . . . , x

] s.t. X = Proj(S /I

), let us recall that X ⊆ P

is arithmetically Cohen-Macaulay (resp. arithmetically Gorenstein) if S /I

is Cohen–Macaulay (resp. Gorenstein).

A classical result by Hartshorne is that aCM schemes are connected in codimension one:

Hartshorne’s connectedness theorem

If X ⊆ P

is aCM, then G (X ) is a connected graph.

On the other hand ...

From graphs to curves

Theorem A, Benedetti-Bolognese-V. 2015

For any connected graph G , there exists a reduced aCM curve C ⊆ P

such that G (C ) = G . Furthermore, reg(C ) = reg(I

) = 3 and the irreducible components of C are rational normal curves no 3 of which meet at one point.

Moreover:

Benedetti-Bolognese-V. 2015

For a connected graph G , the following are equivalent: There is a curve C ⊆ P

such that no 3 of its irreducible components meet at one point, reg(C ) = 2, and G (C ) = G . G is a tree.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

From graphs to curves

Theorem A, Benedetti-Bolognese-V. 2015

For any connected graph G , there exists a reduced aCM curve C ⊆ P

such that G (C ) = G .

Furthermore, reg(C ) = reg(I

) = 3 and the irreducible components of C are rational normal curves no 3 of which meet at one point.

Moreover:

Benedetti-Bolognese-V. 2015

For a connected graph G , the following are equivalent:

There is a curve C ⊆ P

such that no 3 of its irreducible

components meet at one point, reg(C ) = 2, and G (C ) = G .

G is a tree.

From graphs to curves

Theorem A, Benedetti-Bolognese-V. 2015

For any connected graph G , there exists a reduced aCM curve C ⊆ P

such that G (C ) = G . Furthermore, reg(C ) = reg(I

) = 3 and the irreducible components of C are rational normal curves no 3 of which meet at one point.

Moreover:

Benedetti-Bolognese-V. 2015

For a connected graph G , the following are equivalent: There is a curve C ⊆ P

such that no 3 of its irreducible components meet at one point, reg(C ) = 2, and G (C ) = G . G is a tree.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

From graphs to curves

Theorem A, Benedetti-Bolognese-V. 2015

For any connected graph G , there exists a reduced aCM curve C ⊆ P

such that G (C ) = G . Furthermore, reg(C ) = reg(I

) = 3 and the irreducible components of C are rational normal curves no 3 of which meet at one point.

Moreover:

Benedetti-Bolognese-V. 2015

For a connected graph G , the following are equivalent:

There is a curve C ⊆ P

such that no 3 of its irreducible

components meet at one point, reg(C ) = 2, and G (C ) = G .

G is a tree.

From graphs to curves

Theorem A, Benedetti-Bolognese-V. 2015

For any connected graph G , there exists a reduced aCM curve C ⊆ P

such that G (C ) = G . Furthermore, reg(C ) = reg(I

) = 3 and the irreducible components of C are rational normal curves no 3 of which meet at one point.

Moreover:

Benedetti-Bolognese-V. 2015

For a connected graph G , the following are equivalent:

There is a curve C ⊆ P

such that no 3 of its irreducible components meet at one point, reg(C ) = 2, and G (C ) = G .

G is a tree.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

From graphs to curves

Theorem A, Benedetti-Bolognese-V. 2015

For any connected graph G , there exists a reduced aCM curve C ⊆ P

such that G (C ) = G . Furthermore, reg(C ) = reg(I

) = 3 and the irreducible components of C are rational normal curves no 3 of which meet at one point.

Moreover:

Benedetti-Bolognese-V. 2015

For a connected graph G , the following are equivalent:

There is a curve C ⊆ P

such that no 3 of its irreducible

components meet at one point, reg(C ) = 2, and G (C ) = G .

G is a tree.

From graphs to curves

Notice that some graphs cannot be obtained as the dual graph of a line arrangement C , that is a union of lines C = S

i =1

L

. For example, one can see that the graph G having:

{1, . . . , 6} as vertices;

{{i , j} : 1 ≤ i < j ≤ 6} \ {{1, 2}, {3, 4}} as edges is not the dual graph of any line arrangement.

However, by taking 6 generic lines L

⊆ P

and blowing up P

along the points P

= L

∩ L

and P

= L

∩ L

, the strict transform of S

i =1

L

will have G as dual graph!

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

From graphs to curves

Notice that some graphs cannot be obtained as the dual graph of a line arrangement C , that is a union of lines C = S

i =1

L

.

For example, one can see that the graph G having:

{1, . . . , 6} as vertices;

{{i , j} : 1 ≤ i < j ≤ 6} \ {{1, 2}, {3, 4}} as edges is not the dual graph of any line arrangement.

However, by taking 6 generic lines L

⊆ P

and blowing up P

along the points P

= L

∩ L

and P

= L

∩ L

, the strict transform of S

i =1

L

will have G as dual graph!

From graphs to curves

Notice that some graphs cannot be obtained as the dual graph of a line arrangement C , that is a union of lines C = S

i =1

L

. For example, one can see that the graph G having:

{1, . . . , 6} as vertices;

{{i , j} : 1 ≤ i < j ≤ 6} \ {{1, 2}, {3, 4}} as edges

is not the dual graph of any line arrangement.

However, by taking 6 generic lines L

⊆ P

and blowing up P

along the points P

= L

∩ L

and P

= L

∩ L

, the strict transform of S

i =1

L

will have G as dual graph!

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

From graphs to curves

Notice that some graphs cannot be obtained as the dual graph of a line arrangement C , that is a union of lines C = S

i =1

L

. For example, one can see that the graph G having:

{1, . . . , 6} as vertices;

{{i , j} : 1 ≤ i < j ≤ 6} \ {{1, 2}, {3, 4}} as edges is not the dual graph of any line arrangement.

However, by taking 6 generic lines L

⊆ P

and blowing up P

along the points P

= L

∩ L

and P

= L

∩ L

, the strict transform of S

i =1

L

will have G as dual graph!

From graphs to curves

Notice that some graphs cannot be obtained as the dual graph of a line arrangement C , that is a union of lines C = S

i =1

L

. For example, one can see that the graph G having:

{1, . . . , 6} as vertices;

{{i , j} : 1 ≤ i < j ≤ 6} \ {{1, 2}, {3, 4}} as edges is not the dual graph of any line arrangement.

However, by taking 6 generic lines L

⊆ P

and blowing up P

along the points P

= L

∩ L

and P

= L

∩ L

,

the strict transform of S

i =1

L

will have G as dual graph!

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

From graphs to curves

Notice that some graphs cannot be obtained as the dual graph of a line arrangement C , that is a union of lines C = S

i =1

L

. For example, one can see that the graph G having:

{1, . . . , 6} as vertices;

{{i , j} : 1 ≤ i < j ≤ 6} \ {{1, 2}, {3, 4}} as edges is not the dual graph of any line arrangement.

However, by taking 6 generic lines L

⊆ P

and blowing up P

along the points P

= L

∩ L

and P

= L

∩ L

, the strict transform of S

i =1

L

will have G as dual graph!

Connectivity of graphs

The connectedness theorem of Hartshorne says that G (X ) is connected whenever X ⊆ P

is aCM.

We would like to infer something more than connectedness by assuming that X ⊆ P

is arithmetically Gorenstein (e.g. a complete intersection). To this purpose we need to quantify the connectedness of a graph. A graph is d -connected if it has > d vertices, and the deletion of

< d vertices, however chosen, leaves it connected. Menger theorem (Max-flow-min-cut).

A graph is d -connected iff between any 2 vertices one can find d vertex-disjoint paths.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Connectivity of graphs

The connectedness theorem of Hartshorne says that G (X ) is connected whenever X ⊆ P

is aCM. We would like to infer something more than connectedness by assuming that X ⊆ P

is arithmetically Gorenstein (e.g. a complete intersection).

To this purpose we need to quantify the connectedness of a graph. A graph is d -connected if it has > d vertices, and the deletion of

< d vertices, however chosen, leaves it connected. Menger theorem (Max-flow-min-cut).

A graph is d -connected iff between any 2 vertices one can find d

vertex-disjoint paths.

Connectivity of graphs

The connectedness theorem of Hartshorne says that G (X ) is connected whenever X ⊆ P

is aCM. We would like to infer something more than connectedness by assuming that X ⊆ P

is arithmetically Gorenstein (e.g. a complete intersection). To this purpose we need to quantify the connectedness of a graph.

A graph is d -connected if it has > d vertices, and the deletion of

< d vertices, however chosen, leaves it connected. Menger theorem (Max-flow-min-cut).

A graph is d -connected iff between any 2 vertices one can find d vertex-disjoint paths.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Connectivity of graphs

The connectedness theorem of Hartshorne says that G (X ) is connected whenever X ⊆ P

is aCM. We would like to infer something more than connectedness by assuming that X ⊆ P

is arithmetically Gorenstein (e.g. a complete intersection). To this purpose we need to quantify the connectedness of a graph.

A graph is d -connected if it has > d vertices, and the deletion of

< d vertices, however chosen, leaves it connected.

Menger theorem (Max-flow-min-cut).

A graph is d -connected iff between any 2 vertices one can find d

vertex-disjoint paths.

Connectivity of graphs

The connectedness theorem of Hartshorne says that G (X ) is connected whenever X ⊆ P

is aCM. We would like to infer something more than connectedness by assuming that X ⊆ P

is arithmetically Gorenstein (e.g. a complete intersection). To this purpose we need to quantify the connectedness of a graph.

A graph is d -connected if it has > d vertices, and the deletion of

< d vertices, however chosen, leaves it connected.

Menger theorem (Max-flow-min-cut).

A graph is d -connected iff between any 2 vertices one can find d vertex-disjoint paths.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

From schemes to graphs

Theorem B, Benedetti–Bolognese–V. 2015

Let X ⊆ P

be an arithmetically Gorenstein projective scheme such that reg(X ) = reg(I

) = r + 1. If reg(q) ≤ δ for all primary components q of I

, then G (X ) is b(r + δ − 1)/δc-connected. When δ can be chosen to be 1, i.e. when X is a (reduced) union of linear spaces (a subspace arrangement), we recover the following: Benedetti-V. 2014

Let X ⊆ P

be an arithmetically Gorenstein subspace arrangement

such that reg(X ) = reg(I

) = r + 1. Then G (X ) is r -connected.

From schemes to graphs

Theorem B, Benedetti–Bolognese–V. 2015

Let X ⊆ P

be an arithmetically Gorenstein projective scheme such that reg(X ) = reg(I

) = r + 1. If reg(q) ≤ δ for all primary components q of I

, then G (X ) is b(r + δ − 1)/δc-connected.

When δ can be chosen to be 1, i.e. when X is a (reduced) union of linear spaces (a subspace arrangement), we recover the following: Benedetti-V. 2014

Let X ⊆ P

be an arithmetically Gorenstein subspace arrangement such that reg(X ) = reg(I

) = r + 1. Then G (X ) is r -connected.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

From schemes to graphs

Theorem B, Benedetti–Bolognese–V. 2015

Let X ⊆ P

be an arithmetically Gorenstein projective scheme such that reg(X ) = reg(I

) = r + 1. If reg(q) ≤ δ for all primary components q of I

, then G (X ) is b(r + δ − 1)/δc-connected.

When δ can be chosen to be 1, i.e. when X is a (reduced) union of linear spaces (a subspace arrangement), we recover the following:

Benedetti-V. 2014

Let X ⊆ P

be an arithmetically Gorenstein subspace arrangement

such that reg(X ) = reg(I

) = r + 1. Then G (X ) is r -connected.

From schemes to graphs

Theorem B, Benedetti–Bolognese–V. 2015

Let X ⊆ P

be an arithmetically Gorenstein projective scheme such that reg(X ) = reg(I

) = r + 1. If reg(q) ≤ δ for all primary components q of I

, then G (X ) is b(r + δ − 1)/δc-connected.

When δ can be chosen to be 1, i.e. when X is a (reduced) union of linear spaces (a subspace arrangement), we recover the following:

Benedetti-V. 2014

Let X ⊆ P

be an arithmetically Gorenstein subspace arrangement such that reg(X ) = reg(I

) = r + 1. Then G (X ) is r -connected.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Example: 27 lines (revisited)

If a union of 27 lines C ⊆ P

is a complete intersection, that is if I

= (f , g ) ⊆ S = K [x

, x

, x

, x

],

then:

d · e = 27, where d = deg(f ), e = deg(g ). There are 2 cases (up to symmetry) for d and e:

(i)

d = 1 and e = 27, in which case reg(S /I

) = 26.

d = 3 and e = 9, in which case reg(S /I

) = 10. In each case, the previous theorem tells us that G (C ) is

10-connected, which implies that each vertex of G (C ) must have

valency ≥ 10, i.e. each line should meet at least 10 of the others.

Example: 27 lines (revisited)

If a union of 27 lines C ⊆ P

is a complete intersection, that is if I

= (f , g ) ⊆ S = K [x

, x

, x

, x

], then:

d · e = 27, where d = deg(f ), e = deg(g ).

There are 2 cases (up to symmetry) for d and e:

d = 1 and e = 27, in which case reg(S /I

) = 26.

d = 3 and e = 9, in which case reg(S /I

) = 10. In each case, the previous theorem tells us that G (C ) is

10-connected, which implies that each vertex of G (C ) must have valency ≥ 10, i.e. each line should meet at least 10 of the others.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Example: 27 lines (revisited)

If a union of 27 lines C ⊆ P

is a complete intersection, that is if I

= (f , g ) ⊆ S = K [x

, x

, x

, x

], then:

d · e = 27, where d = deg(f ), e = deg(g ).

There are 2 cases (up to symmetry) for d and e:

(i)

d = 1 and e = 27, in which case reg(S /I

) = 26.

d = 3 and e = 9, in which case reg(S /I

) = 10. In each case, the previous theorem tells us that G (C ) is

10-connected, which implies that each vertex of G (C ) must have

valency ≥ 10, i.e. each line should meet at least 10 of the others.

Example: 27 lines (revisited)

If a union of 27 lines C ⊆ P

is a complete intersection, that is if I

= (f , g ) ⊆ S = K [x

, x

, x

, x

], then:

d · e = 27, where d = deg(f ), e = deg(g ).

There are 2 cases (up to symmetry) for d and e:

(i)

d = 1 and e = 27, in which case reg(S /I

) = 26.

(ii)

d = 3 and e = 9, in which case reg(S /I

) = 10. In each case, the previous theorem tells us that G (C ) is

10-connected, which implies that each vertex of G (C ) must have valency ≥ 10, i.e. each line should meet at least 10 of the others.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Example: 27 lines (revisited)

If a union of 27 lines C ⊆ P

is a complete intersection, that is if I

= (f , g ) ⊆ S = K [x

, x

, x

, x

], then:

d · e = 27, where d = deg(f ), e = deg(g ).

There are 2 cases (up to symmetry) for d and e:

(i)

d = 1 and e = 27, in which case reg(S /I

) = 26.

(ii)

d = 3 and e = 9, in which case reg(S /I

) = 10.

In each case, the previous theorem tells us that G (C ) is

10-connected, which implies that each vertex of G (C ) must have

valency ≥ 10, i.e. each line should meet at least 10 of the others.

Example: 27 lines (revisited)

If a union of 27 lines C ⊆ P

is a complete intersection, that is if I

= (f , g ) ⊆ S = K [x

, x

, x

, x

], then:

d · e = 27, where d = deg(f ), e = deg(g ).

There are 2 cases (up to symmetry) for d and e:

(i)

d = 1 and e = 27, in which case reg(S /I

) = 26.

(ii)

d = 3 and e = 9, in which case reg(S /I

) = 10.

In each case, the previous theorem tells us that G (C ) is 10-connected,

which implies that each vertex of G (C ) must have valency ≥ 10, i.e. each line should meet at least 10 of the others.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Example: 27 lines (revisited)

If a union of 27 lines C ⊆ P

is a complete intersection, that is if I

= (f , g ) ⊆ S = K [x

, x

, x

, x

], then:

d · e = 27, where d = deg(f ), e = deg(g ).

There are 2 cases (up to symmetry) for d and e:

(i)

d = 1 and e = 27, in which case reg(S /I

) = 26.

(ii)

d = 3 and e = 9, in which case reg(S /I

) = 10.

In each case, the previous theorem tells us that G (C ) is

10-connected, which implies that each vertex of G (C ) must have valency ≥ 10,

i.e. each line should meet at least 10 of the others.

Example: 27 lines (revisited)

If a union of 27 lines C ⊆ P

is a complete intersection, that is if I

= (f , g ) ⊆ S = K [x

, x

, x

, x

], then:

d · e = 27, where d = deg(f ), e = deg(g ).

There are 2 cases (up to symmetry) for d and e:

(i)

d = 1 and e = 27, in which case reg(S /I

) = 26.

(ii)

d = 3 and e = 9, in which case reg(S /I

) = 10.

In each case, the previous theorem tells us that G (C ) is

10-connected, which implies that each vertex of G (C ) must have valency ≥ 10, i.e. each line should meet at least 10 of the others.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Line arrangements in P

Another way to quantify the connectedness of a graph is by meaning of its diameter, that is the maximum distance between 2 of its vertices.

One can verify that the 27 lines lying on a smooth cubic have diameter 2. In fact, I do not know any example of aCM line arrangement C ⊆ P

such that diam(G (C )) > 2.

The following experiments are due to Michela Di Marca: pick

`

, . . . , `

linear forms in K [x

, x

, x

, x

] any 4 of which are linearly

independent. Let G be a graph on N + 1 vertices such that its dual

has diameter k. In this way, the dual graph of the line arrangement

C = T{`

= `

= 0}, where {i , j } is an edge of G , has diameter k.

She ran lots of examples choosing `

= x

+ ix

+ i

x

+ i

x

(or

other variations), but in all tested cases C aCM =⇒ k ≤ 2.

Line arrangements in P

Another way to quantify the connectedness of a graph is by meaning of its diameter, that is the maximum distance between 2 of its vertices. One can verify that the 27 lines lying on a smooth cubic have diameter 2.

In fact, I do not know any example of aCM line arrangement C ⊆ P

such that diam(G (C )) > 2.

The following experiments are due to Michela Di Marca: pick

`

, . . . , `

linear forms in K [x

, x

, x

, x

] any 4 of which are linearly independent. Let G be a graph on N + 1 vertices such that its dual has diameter k. In this way, the dual graph of the line arrangement C = T{`

= `

= 0}, where {i , j } is an edge of G , has diameter k. She ran lots of examples choosing `

= x

+ ix

+ i

x

+ i

x

(or other variations), but in all tested cases C aCM =⇒ k ≤ 2.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Line arrangements in P

Another way to quantify the connectedness of a graph is by meaning of its diameter, that is the maximum distance between 2 of its vertices. One can verify that the 27 lines lying on a smooth cubic have diameter 2. In fact, I do not know any example of aCM line arrangement C ⊆ P

such that diam(G (C )) > 2.

The following experiments are due to Michela Di Marca: pick

`

, . . . , `

linear forms in K [x

, x

, x

, x

] any 4 of which are linearly

independent. Let G be a graph on N + 1 vertices such that its dual

has diameter k. In this way, the dual graph of the line arrangement

C = T{`

= `

= 0}, where {i , j } is an edge of G , has diameter k.

She ran lots of examples choosing `

= x

+ ix

+ i

x

+ i

x

(or

other variations), but in all tested cases C aCM =⇒ k ≤ 2.

Line arrangements in P

Another way to quantify the connectedness of a graph is by meaning of its diameter, that is the maximum distance between 2 of its vertices. One can verify that the 27 lines lying on a smooth cubic have diameter 2. In fact, I do not know any example of aCM line arrangement C ⊆ P

such that diam(G (C )) > 2.

The following experiments are due to Michela Di Marca:

pick

`

, . . . , `

linear forms in K [x

, x

, x

, x

] any 4 of which are linearly independent. Let G be a graph on N + 1 vertices such that its dual has diameter k. In this way, the dual graph of the line arrangement C = T{`

= `

= 0}, where {i , j } is an edge of G , has diameter k. She ran lots of examples choosing `

= x

+ ix

+ i

x

+ i

x

(or other variations), but in all tested cases C aCM =⇒ k ≤ 2.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Line arrangements in P

Another way to quantify the connectedness of a graph is by meaning of its diameter, that is the maximum distance between 2 of its vertices. One can verify that the 27 lines lying on a smooth cubic have diameter 2. In fact, I do not know any example of aCM line arrangement C ⊆ P

such that diam(G (C )) > 2.

The following experiments are due to Michela Di Marca: pick

`

, . . . , `

linear forms in K [x

, x

, x

, x

] any 4 of which are linearly independent.

Let G be a graph on N + 1 vertices such that its dual

has diameter k. In this way, the dual graph of the line arrangement

C = T{`

= `

= 0}, where {i , j } is an edge of G , has diameter k.

She ran lots of examples choosing `

= x

+ ix

+ i

x

+ i

x

(or

other variations), but in all tested cases C aCM =⇒ k ≤ 2.

Line arrangements in P

Another way to quantify the connectedness of a graph is by meaning of its diameter, that is the maximum distance between 2 of its vertices. One can verify that the 27 lines lying on a smooth cubic have diameter 2. In fact, I do not know any example of aCM line arrangement C ⊆ P

such that diam(G (C )) > 2.

The following experiments are due to Michela Di Marca: pick

`

, . . . , `

linear forms in K [x

, x

, x

, x

] any 4 of which are linearly independent. Let G be a graph on N + 1 vertices such that its dual has diameter k.

In this way, the dual graph of the line arrangement C = T{`

= `

= 0}, where {i , j } is an edge of G , has diameter k. She ran lots of examples choosing `

= x

+ ix

+ i

x

+ i

x

(or other variations), but in all tested cases C aCM =⇒ k ≤ 2.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Line arrangements in P

Another way to quantify the connectedness of a graph is by meaning of its diameter, that is the maximum distance between 2 of its vertices. One can verify that the 27 lines lying on a smooth cubic have diameter 2. In fact, I do not know any example of aCM line arrangement C ⊆ P

such that diam(G (C )) > 2.

The following experiments are due to Michela Di Marca: pick

`

, . . . , `

linear forms in K [x

, x

, x

, x

] any 4 of which are linearly independent. Let G be a graph on N + 1 vertices such that its dual has diameter k. In this way, the dual graph of the line arrangement C = T{`

= `

= 0}, where {i , j } is an edge of G , has diameter k.

She ran lots of examples choosing `

= x

+ ix

+ i

x

+ i

x

(or

other variations), but in all tested cases C aCM =⇒ k ≤ 2.

Line arrangements in P

Another way to quantify the connectedness of a graph is by meaning of its diameter, that is the maximum distance between 2 of its vertices. One can verify that the 27 lines lying on a smooth cubic have diameter 2. In fact, I do not know any example of aCM line arrangement C ⊆ P

such that diam(G (C )) > 2.

The following experiments are due to Michela Di Marca: pick

`

, . . . , `

linear forms in K [x

, x

, x

, x

] any 4 of which are linearly independent. Let G be a graph on N + 1 vertices such that its dual has diameter k. In this way, the dual graph of the line arrangement C = T{`

= `

= 0}, where {i , j } is an edge of G , has diameter k.

She ran lots of examples choosing `

= x

+ ix

+ i

x

+ i

x

(or other variations), but in all tested cases C aCM =⇒ k ≤ 2.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Hirsch embeddings

We say that a projective scheme X ⊆ P

is Hirsch if diam(G (X )) ≤ codim

X .

What said in the previous slide suggests the following: Question

Is any aCM line arrangement C ⊆ P

Hirsch? Be careful:

There exist nonreduced complete intersections C ⊆ P

such that C

⊆ P

is a line arrangement and diam(G (C )) is arbitrarily large.

For large n, there are arithmetically Gorenstein line

arrangements in P

that are not Hirsch (Santos).

Hirsch embeddings

We say that a projective scheme X ⊆ P

is Hirsch if diam(G (X )) ≤ codim

X .

What said in the previous slide suggests the following: Question

Is any aCM line arrangement C ⊆ P

Hirsch? Be careful:

There exist nonreduced complete intersections C ⊆ P

such that C

⊆ P

is a line arrangement and diam(G (C )) is arbitrarily large.

For large n, there are arithmetically Gorenstein line arrangements in P

that are not Hirsch (Santos).

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Hirsch embeddings

We say that a projective scheme X ⊆ P

is Hirsch if diam(G (X )) ≤ codim

X .

What said in the previous slide suggests the following:

Question

Is any aCM line arrangement C ⊆ P

Hirsch?

Be careful:

There exist nonreduced complete intersections C ⊆ P

such that C

⊆ P

is a line arrangement and diam(G (C )) is arbitrarily large.

For large n, there are arithmetically Gorenstein line

arrangements in P

that are not Hirsch (Santos).

Hirsch embeddings

We say that a projective scheme X ⊆ P

is Hirsch if diam(G (X )) ≤ codim

X .

What said in the previous slide suggests the following:

Question

Is any aCM line arrangement C ⊆ P

Hirsch?

Be careful:

There exist nonreduced complete intersections C ⊆ P

such that C

⊆ P

is a line arrangement and diam(G (C )) is arbitrarily large.

For large n, there are arithmetically Gorenstein line arrangements in P

that are not Hirsch (Santos).

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Hirsch embeddings

Many projective embeddings, however, are Hirsch:

Adiprasito–Benedetti 2014

If X ⊆ P

is aCM and I

is a monomial ideal generated by quadrics, then X ⊆ P

is Hirsch.

Benedetti–V. 2014

If X is an arrangement of lines, no 3 of which meet in the same point, canonically embedded in P

, then X ⊆ P

is Hirsch. Conjecture: Benedetti–V. 2014

If X ⊆ P

is a (reduced) aCM scheme and I

is generated by

quadrics, then X ⊆ P

is Hirsch.

Hirsch embeddings

Many projective embeddings, however, are Hirsch:

Adiprasito–Benedetti 2014

If X ⊆ P

is aCM and I

is a monomial ideal generated by quadrics, then X ⊆ P

is Hirsch.

Benedetti–V. 2014

If X is an arrangement of lines, no 3 of which meet in the same point, canonically embedded in P

, then X ⊆ P

is Hirsch.

Conjecture: Benedetti–V. 2014

If X ⊆ P

is a (reduced) aCM scheme and I

is generated by quadrics, then X ⊆ P

is Hirsch.

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Hirsch embeddings

Many projective embeddings, however, are Hirsch:

Adiprasito–Benedetti 2014

If X ⊆ P

is aCM and I

is a monomial ideal generated by quadrics, then X ⊆ P

is Hirsch.

Benedetti–V. 2014

If X is an arrangement of lines, no 3 of which meet in the same point, canonically embedded in P

, then X ⊆ P

is Hirsch.

Conjecture: Benedetti–V. 2014

If X ⊆ P

is a (reduced) aCM scheme and I

is generated by

quadrics, then X ⊆ P

is Hirsch.

Something needed to prove Theorem B

First of all, by taking generic hyperplane sections one can reduce himself to consider dim X = 1. At this point one has to use liaison theory in a clever way... The following is an important ingredient: Essentially Caviglia 2007

If I = ∩

q

is a primary decomposition of a homogeneous ideal I ⊆ S = K [x

, . . . , x

] and Proj(S /I ) has dimension 1, then:

reg(I ) ≤

s

X

i =1

reg(q

).

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Something needed to prove Theorem B

First of all, by taking generic hyperplane sections one can reduce himself to consider dim X = 1.

At this point one has to use liaison theory in a clever way... The following is an important ingredient: Essentially Caviglia 2007

If I = ∩

q

is a primary decomposition of a homogeneous ideal I ⊆ S = K [x

, . . . , x

] and Proj(S /I ) has dimension 1, then:

reg(I ) ≤

s

X

i =1

reg(q

).

Something needed to prove Theorem B

First of all, by taking generic hyperplane sections one can reduce himself to consider dim X = 1. At this point one has to use liaison theory in a clever way...

The following is an important ingredient: Essentially Caviglia 2007

If I = ∩

q

is a primary decomposition of a homogeneous ideal I ⊆ S = K [x

, . . . , x

] and Proj(S /I ) has dimension 1, then:

reg(I ) ≤

s

X

i =1

reg(q

).

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Something needed to prove Theorem B

First of all, by taking generic hyperplane sections one can reduce himself to consider dim X = 1. At this point one has to use liaison theory in a clever way... The following is an important ingredient:

Essentially Caviglia 2007

If I = ∩

q

is a primary decomposition of a homogeneous ideal I ⊆ S = K [x

, . . . , x

] and Proj(S /I ) has dimension 1, then:

reg(I ) ≤

s

X

i =1

reg(q

).

Something needed to prove Theorem B

First of all, by taking generic hyperplane sections one can reduce himself to consider dim X = 1. At this point one has to use liaison theory in a clever way... The following is an important ingredient:

Essentially Caviglia 2007

If I = ∩

q

is a primary decomposition of a homogeneous ideal I ⊆ S = K [x

, . . . , x

] and Proj(S /I ) has dimension 1, then:

reg(I ) ≤

s

X

i =1

reg(q

).

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

Something needed to prove Theorem B

First of all, by taking generic hyperplane sections one can reduce himself to consider dim X = 1. At this point one has to use liaison theory in a clever way... The following is an important ingredient:

Essentially Caviglia 2007

If I = ∩

q

is a primary decomposition of a homogeneous ideal I ⊆ S = K [x

, . . . , x

] and Proj(S /I ) has dimension 1, then:

reg(I ) ≤

s

X

i =1

reg(q

).

An ’Eisenbud-Goto style’ question

Eisenbud-Goto conjecture (1984)

Let X ⊆ P

be a nondegenerate reduced projective scheme with connected dual graph. Then

reg(X ) ≤ deg(X ) − codim

X + 1. The conjecture is known to be true in its full generality in dimension 1 by Gruson-Lazarsfeld-Peskine and Giaimo; in dimension 2, it is true for smooth surfaces by Lazarsfeld; for smooth threefolds and fourfolds, it is ’almost’ true by Kwak. By the subadditivity result of Caviglia, the EG for curves yields: Theorem

Let C ⊆ P

be an equidimensional reduced projective curve. Then reg(C ) ≤ deg(C )

Matteo Varbaro (University of Genova) Dual graphs of projective schemes

An ’Eisenbud-Goto style’ question

Eisenbud-Goto conjecture (1984)

Let X ⊆ P

be a nondegenerate reduced projective scheme with connected dual graph. Then

reg(X ) ≤ deg(X ) − codim

X + 1.

The conjecture is known to be true in its full generality in dimension 1 by Gruson-Lazarsfeld-Peskine and Giaimo; in dimension 2, it is true for smooth surfaces by Lazarsfeld; for smooth threefolds and fourfolds, it is ’almost’ true by Kwak. By the subadditivity result of Caviglia, the EG for curves yields: Theorem

Let C ⊆ P

be an equidimensional reduced projective curve. Then

reg(C ) ≤ deg(C )