Cohomological Dimension, Symbolic Powers and Matroids

Cohomological Dimension, Symbolic Powers and Matroids

Matteo Varbaro

Dipartimento di Matematica, Universit`a di Genova

April 4, 2011

Matteo Varbaro, Universit`a di Genova PhD thesis

PAPERS AND STRUCTURE OF THE THESIS

1 L. Sharifan, M. Varbaro, Graded Betti numbers and ideals with linear quotients, Le Matematiche, Vol. LXIII (2008), pp. 257-265.

2 B. Benedetti, A. Constantinescu, M. Varbaro, Dimension, depth and zero-divisors of the algebra of basic k-covers, Le Matematiche, Vol. LXIII (2008), pp. 117-156.

3 M. Varbaro, Gr¨obner deformations, connectedness and cohomological dimension, Journal of Algebra, Vol. 322 (2009), pp. 2492-2507.

4 B. Benedetti, M. Varbaro, Unmixed graphs that are domains, to appear in Communications in Algebra.

5 M. Varbaro, Arithmetical rank of certain Segre embeddings, to appear in Transactions of the American Mathematical Society.

6 A. Constantinescu, M. Varbaro, Koszulness, Krull dimension and other properties of graph-related algebras, to appear in Journal of Algebraic Combinatorics.

7 M. Varbaro, Symbolic powers and matroids, to appear in Proceedings of the American Mathematical Society.

8 L. D. Nam, M. Varbaro, Cohen-Macaulayness of generically complete intersection monomial ideals, to appear in Communications in Algebra.

9 A. Constantinescu, M. Varbaro, On the h-vectors of Cohen-Macaulay Flag Complexes, submitted.

10 W. Bruns, A. Conca, M. Varbaro, Relations among the Minors of a Generic Matrix, in preparation.

Matteo Varbaro, Universit`a di Genova PhD thesis

PAPERS AND STRUCTURE OF THE THESIS

1 L. Sharifan, M. Varbaro, Graded Betti numbers and ideals with linear quotients, Le Matematiche, Vol. LXIII (2008), pp. 257-265.

2 B. Benedetti, A. Constantinescu, M. Varbaro, Dimension, depth and zero-divisors of the algebra of basic k-covers, Le Matematiche, Vol. LXIII (2008), pp. 117-156.

3 M. Varbaro, Gr¨obner deformations, connectedness and cohomological dimension, Journal of Algebra, Vol. 322 (2009), pp. 2492-2507.

4 B. Benedetti, M. Varbaro, Unmixed graphs that are domains, to appear in Communications in Algebra.

5 M. Varbaro, Arithmetical rank of certain Segre embeddings, to appear in Transactions of the American Mathematical Society.

6 A. Constantinescu, M. Varbaro, Koszulness, Krull dimension and other properties of graph-related algebras, to appear in Journal of Algebraic Combinatorics.

7 M. Varbaro, Symbolic powers and matroids, to appear in Proceedings of the American Mathematical Society.

8 L. D. Nam, M. Varbaro, Cohen-Macaulayness of generically complete intersection monomial ideals, to appear in Communications in Algebra.

9 A. Constantinescu, M. Varbaro, On the h-vectors of Cohen-Macaulay Flag Complexes, submitted.

10 W. Bruns, A. Conca, M. Varbaro, Relations among the Minors of a Generic Matrix, in preparation.

Matteo Varbaro, Universit`a di Genova PhD thesis

PAPERS AND STRUCTURE OF THE THESIS

1 L. Sharifan, M. Varbaro, Graded Betti numbers and ideals with linear quotients, Le Matematiche, Vol. LXIII (2008), pp. 257-265.

2 B. Benedetti, A. Constantinescu, M. Varbaro, Dimension, depth and zero-divisors of the algebra of basic k-covers, Le Matematiche, Vol. LXIII (2008), pp. 117-156.

3 M. Varbaro, Gr¨obner deformations, connectedness and cohomological dimension, Journal of Algebra, Vol. 322 (2009), pp. 2492-2507.

4 B. Benedetti, M. Varbaro, Unmixed graphs that are domains, to appear in Communications in Algebra.

5 M. Varbaro, Arithmetical rank of certain Segre embeddings, to appear in Transactions of the American Mathematical Society.

6 A. Constantinescu, M. Varbaro, Koszulness, Krull dimension and other properties of graph-related algebras, to appear in Journal of Algebraic Combinatorics.

7 M. Varbaro, Symbolic powers and matroids, to appear in Proceedings of the American Mathematical Society.

8 L. D. Nam, M. Varbaro, Cohen-Macaulayness of generically complete intersection monomial ideals, to appear in Communications in Algebra.

9 A. Constantinescu, M. Varbaro, On the h-vectors of Cohen-Macaulay Flag Complexes, submitted.

10 W. Bruns, A. Conca, M. Varbaro, Relations among the Minors of a Generic Matrix, in preparation.

Matteo Varbaro, Universit`a di Genova PhD thesis

PAPERS AND STRUCTURE OF THE THESIS

1 L. Sharifan, M. Varbaro, Graded Betti numbers and ideals with linear quotients, Le Matematiche, Vol. LXIII (2008), pp. 257-265.

2 B. Benedetti, A. Constantinescu, M. Varbaro, Dimension, depth and zero-divisors of the algebra of basic k-covers, Le Matematiche, Vol. LXIII (2008), pp. 117-156.

3 M. Varbaro, Gr¨obner deformations, connectedness and cohomological dimension, Journal of Algebra, Vol. 322 (2009), pp. 2492-2507.

4 B. Benedetti, M. Varbaro, Unmixed graphs that are domains, to appear in Communications in Algebra.

5 M. Varbaro, Arithmetical rank of certain Segre embeddings, to appear in Transactions of the American Mathematical Society.

6 A. Constantinescu, M. Varbaro, Koszulness, Krull dimension and other properties of graph-related algebras, to appear in Journal of Algebraic Combinatorics.

7 M. Varbaro, Symbolic powers and matroids, to appear in Proceedings of the American Mathematical Society.

8 L. D. Nam, M. Varbaro, Cohen-Macaulayness of generically complete intersection monomial ideals, to appear in Communications in Algebra.

9 A. Constantinescu, M. Varbaro, On the h-vectors of Cohen-Macaulay Flag Complexes, submitted.

10 W. Bruns, A. Conca, M. Varbaro, Relations among the Minors of a Generic Matrix, in preparation.

Matteo Varbaro, Universit`a di Genova PhD thesis

PAPERS AND STRUCTURE OF THE THESIS

1 L. Sharifan, M. Varbaro, Graded Betti numbers and ideals with linear quotients, Le Matematiche, Vol. LXIII (2008), pp. 257-265.

2 B. Benedetti, A. Constantinescu, M. Varbaro, Dimension, depth and zero-divisors of the algebra of basic k-covers, Le Matematiche, Vol. LXIII (2008), pp. 117-156.

3 M. Varbaro, Gr¨obner deformations, connectedness and cohomological dimension, Journal of Algebra, Vol. 322 (2009), pp. 2492-2507.

4 B. Benedetti, M. Varbaro, Unmixed graphs that are domains, to appear in Communications in Algebra.

5 M. Varbaro, Arithmetical rank of certain Segre embeddings, to appear in Transactions of the American Mathematical Society.

6 A. Constantinescu, M. Varbaro, Koszulness, Krull dimension and other properties of graph-related algebras, to appear in Journal of Algebraic Combinatorics.

7 M. Varbaro, Symbolic powers and matroids, to appear in Proceedings of the American Mathematical Society.

8 L. D. Nam, M. Varbaro, Cohen-Macaulayness of generically complete intersection monomial ideals, to appear in Communications in Algebra.

9 A. Constantinescu, M. Varbaro, On the h-vectors of Cohen-Macaulay Flag Complexes, submitted.

10 W. Bruns, A. Conca, M. Varbaro, Relations among the Minors of a Generic Matrix, in preparation.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY

VS

ETALE TOPOLOGY´

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

T h e a r i t h m e t i c a l r a n k

LetS := k[x0, . . . , x_n] be the polynomial ring in n + 1 variables over an algebraically closed field k,f₁, . . . , f_r homogeneous polynomials of S and

X := {P ∈ Pⁿ: f1(P) = . . . = fr(P) = 0} ⊆ Pⁿ.

QUESTION: Which is the minimum number of homogeneous polynomials which define X ?

ara(X ) := minimum s ∈ N: ∃ g1, . . . , g_s ∈ S homogeneous with X = {P ∈ Pⁿ: g₁(P) = . . . = g_s(P) = 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

T h e a r i t h m e t i c a l r a n k

LetS := k[x0, . . . , x_n] be the polynomial ring inn + 1 variables over an algebraically closed fieldk,f₁, . . . , f_r homogeneous polynomials of S and

X := {P ∈ Pⁿ: f1(P) = . . . = fr(P) = 0} ⊆ Pⁿ.

QUESTION: Which is the minimum number of homogeneous polynomials which define X ?

ara(X ) := minimum s ∈ N: ∃ g1, . . . , g_s ∈ S homogeneous with X = {P ∈ Pⁿ: g₁(P) = . . . = g_s(P) = 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

T h e a r i t h m e t i c a l r a n k

LetS := k[x0, . . . , x_n] be the polynomial ring in n + 1 variables over an algebraically closed field k,f₁, . . . , f_r homogeneous polynomials of S and

X := {P ∈ Pⁿ: f1(P) = . . . = fr(P) = 0} ⊆ Pⁿ.

QUESTION: Which is the minimum number of homogeneous polynomials which define X ?

ara(X ) := minimum s ∈ N: ∃ g1, . . . , g_s ∈ S homogeneous with X = {P ∈ Pⁿ: g₁(P) = . . . = g_s(P) = 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

T h e a r i t h m e t i c a l r a n k

LetS := k[x0, . . . , x_n] be the polynomial ring in n + 1 variables over an algebraically closed field k,f₁, . . . , f_r homogeneous polynomials of S and

X := {P ∈Pⁿ: f1(P) = . . . = fr(P) = 0} ⊆ Pⁿ.

QUESTION: Which is the minimum number of homogeneous polynomials which define X ?

ara(X ):= minimum s ∈ N: ∃ g1, . . . , g_s ∈ S homogeneous with X = {P ∈ Pⁿ: g₁(P) = . . . = g_s(P) = 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

T h e a r i t h m e t i c a l r a n k

LetS := k[x0, . . . , x_n] be the polynomial ring in n + 1 variables over an algebraically closed field k,f₁, . . . , f_r homogeneous polynomials of S and

X := {P ∈ Pⁿ: f1(P) = . . . = fr(P) = 0} ⊆ Pⁿ.

QUESTION: Which is the minimum number of homogeneous polynomials which define X ?

ara(X ):= minimum s ∈ N: ∃ g1, . . . , g_s ∈ S homogeneous with X = {P ∈ Pⁿ: g₁(P) = . . . = g_s(P) = 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

T h e a r i t h m e t i c a l r a n k

LetS := k[x0, . . . , x_n] be the polynomial ring in n + 1 variables over an algebraically closed field k,f₁, . . . , f_r homogeneous polynomials of S and

X := {P ∈ Pⁿ: f1(P) = . . . = fr(P) = 0} ⊆ Pⁿ.

QUESTION: Which is the minimum number of homogeneous polynomials which define X ?

ara(X ):= minimum s ∈ N: ∃ g1, . . . , g_s ∈ S homogeneous with X = {P ∈ Pⁿ: g₁(P) = . . . = g_s(P) = 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

T h e a r i t h m e t i c a l r a n k

LetS := k[x0, . . . , x_n] be the polynomial ring in n + 1 variables over an algebraically closed field k,f₁, . . . , f_r homogeneous polynomials of S and

X := {P ∈ Pⁿ: f1(P) = . . . = fr(P) = 0} ⊆ Pⁿ.

QUESTION: Which is the minimum number of homogeneous polynomials which define X ?

ara(X ):= minimum s ∈ N: ∃ g1, . . . , g_s ∈ S homogeneous with X = {P ∈ Pⁿ: g₁(P) = . . . = g_s(P) = 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Eisenbud, Evans, 1972): X 6= ∅ =⇒ ara(X ) ≤ n.

(Krull, 1938): ara(X ) ≥ codim_Pⁿ(X ).

(Serre, 1955):

(Grothendieck, 1973): .

For any scheme U, we define the cohomological dimension of U as:

cd(U) := sup{i ∈ N : ∃ F quasi-coherent : Hⁱ(U, F ) 6= 0}.

The ´etale cohomological dimension of U is

´ecd(U) := sup{i ∈ N : ∃ G `-torsion : Hⁱ(U_´et, G) 6= 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Eisenbud, Evans, 1972): X 6= ∅ =⇒ ara(X ) ≤ n.

(Krull, 1938): ara(X ) ≥ codim_Pⁿ(X ).

(Serre, 1955):

(Grothendieck, 1973): .

For any scheme U, we define the cohomological dimension of U as:

cd(U) := sup{i ∈ N : ∃ F quasi-coherent : Hⁱ(U, F ) 6= 0}.

The ´etale cohomological dimension of U is

´ecd(U) := sup{i ∈ N : ∃ G `-torsion : Hⁱ(U_´et, G) 6= 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Eisenbud, Evans, 1972): X 6= ∅ =⇒ ara(X ) ≤ n.

(Krull, 1938): ara(X ) ≥ codim_Pⁿ(X ).

(Serre, 1955):

(Grothendieck, 1973): .

For any scheme U, we define the cohomological dimension of U as:

cd(U) := sup{i ∈ N : ∃ F quasi-coherent : Hⁱ(U, F ) 6= 0}.

The ´etale cohomological dimension of U is

´ecd(U) := sup{i ∈ N : ∃ G `-torsion : Hⁱ(U_´et, G) 6= 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Eisenbud, Evans, 1972): X 6= ∅ =⇒ ara(X ) ≤ n.

(Krull, 1938): ara(X ) ≥ codim_Pⁿ(X ).

(Serre, 1955): Hⁱ(Pⁿ\ X , F ) = 0∀ i ≥ ara(X ).

(Grothendieck, 1973): .

For any scheme U, we define thecohomological dimension of U as:

cd(U):= sup{i ∈ N : ∃ F quasi-coherent : Hⁱ(U, F ) 6= 0}.

The ´etale cohomological dimension of U is

´ecd(U) := sup{i ∈ N : ∃ G `-torsion : Hⁱ(U_´et, G) 6= 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Eisenbud, Evans, 1972): X 6= ∅ =⇒ ara(X ) ≤ n.

(Krull, 1938): ara(X ) ≥ codim_Pⁿ(X ).

(Serre, 1955): Hⁱ(Pⁿ\ X , F ) = 0∀ i ≥ ara(X ).

(Grothendieck, 1973): Hⁱ((Pⁿ\ X )_´et, G) = 0 ∀ i ≥ ara(X ) + n.

For any scheme U, we define thecohomological dimension of U as:

cd(U):= sup{i ∈ N : ∃ F quasi-coherent : Hⁱ(U, F ) 6= 0}.

The ´etale cohomological dimension of U is

´ecd(U) := sup{i ∈ N : ∃ G `-torsion : Hⁱ(U_´et, G) 6= 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Eisenbud, Evans, 1972): X 6= ∅ =⇒ ara(X ) ≤ n.

(Krull, 1938): ara(X ) ≥ codim_Pⁿ(X ).

(Serre, 1955): Hⁱ(Pⁿ\ X , F ) = 0∀ i ≥ ara(X ).

(Grothendieck, 1973): Hⁱ((Pⁿ\ X )_´et, G) = 0 ∀ i ≥ ara(X ) + n.

For any schemeU, we define the cohomological dimension of U as:

cd(U):= sup{i ∈ N : ∃ F quasi-coherent : Hⁱ(U, F ) 6= 0}.

The´etale cohomological dimension of U is

´

ecd(U):= sup{i ∈ N : ∃ G `-torsion : Hⁱ(U_´et, G) 6= 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Eisenbud, Evans, 1972): X 6= ∅ =⇒ ara(X ) ≤ n.

(Krull, 1938): ara(X ) ≥ codim_Pⁿ(X ).

(Serre, 1955): ara(X ) ≥ cd(Pⁿ\ X ) + 1.

(Grothendieck, 1973): Hⁱ((Pⁿ\ X )_´et, G) = 0 ∀ i ≥ ara(X ) + n.

For any scheme U, we define thecohomological dimension of U as:

cd(U):= sup{i ∈ N : ∃ F quasi-coherent : Hⁱ(U, F ) 6= 0}.

The´etale cohomological dimension of U is

´

ecd(U):= sup{i ∈ N : ∃ G `-torsion : Hⁱ(U_´et, G) 6= 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Eisenbud, Evans, 1972): X 6= ∅ =⇒ ara(X ) ≤ n.

(Krull, 1938): ara(X ) ≥ codim_Pⁿ(X ).

(Serre, 1955): ara(X ) ≥ cd(Pⁿ\ X ) + 1.

(Grothendieck, 1973): Hⁱ((Pⁿ\ X )_´et, G) = 0 ∀ i ≥ ara(X ) + n.

For any scheme U, we define thecohomological dimension of U as:

cd(U):= sup{i ∈ N : ∃ F quasi-coherent : Hⁱ(U, F ) 6= 0}.

The´etale cohomological dimension of U is

´

ecd(U):= sup{i ∈ N : ∃ G `-torsion : Hⁱ(U_´et, G) 6= 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Eisenbud, Evans, 1972): X 6= ∅ =⇒ ara(X ) ≤ n.

(Krull, 1938): ara(X ) ≥ codim_Pⁿ(X ).

(Serre, 1955): ara(X ) ≥ cd(Pⁿ\ X ) + 1.

(Grothendieck, 1973): ara(X ) ≥ ´ecd(Pⁿ\ X ) − n + 1.

For any scheme U, we define thecohomological dimension of U as:

cd(U):= sup{i ∈ N : ∃ F quasi-coherent : Hⁱ(U, F ) 6= 0}.

The´etale cohomological dimension of U is

´

ecd(U):= sup{i ∈ N : ∃ G `-torsion : Hⁱ(U_´et, G) 6= 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Eisenbud, Evans, 1972): X 6= ∅ =⇒ ara(X ) ≤ n.

(Krull, 1938): ara(X ) ≥ codim_Pⁿ(X ).

(Serre, 1955): ara(X ) ≥ cd(Pⁿ\ X ) + 1.

(Grothendieck, 1973): ara(X ) ≥ ´ecd(Pⁿ\ X ) − n + 1.

For any scheme U, we define thecohomological dimension of U as:

cd(U):= sup{i ∈ N : ∃ F quasi-coherent : Hⁱ(U, F ) 6= 0}.

The´etale cohomological dimension of U is

´

ecd(U):= sup{i ∈ N : ∃ G `-torsion : Hⁱ(U_´et, G) 6= 0}.

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Grothendieck, 1961): cd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) − 1.

(Lyubeznik, 1993): ´ecd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) + n − 1.

As we learned in the above slide:

ara(X ) ≥ cd(Pⁿ\ X ) + 1.

ara(X ) ≥ ´ecd(Pⁿ\ X ) − n + 1.

QUESTION (Hartshorne, 1970): Which is better???

?

´ecd(Pⁿ\ X ) ≷ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Grothendieck, 1961): cd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) − 1.

(Lyubeznik, 1993): ´ecd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) + n − 1.

As we learned in the above slide:

ara(X ) ≥ cd(Pⁿ\ X ) + 1.

ara(X ) ≥ ´ecd(Pⁿ\ X ) − n + 1.

QUESTION (Hartshorne, 1970): Which is better???

?

´ecd(Pⁿ\ X ) ≷ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Grothendieck, 1961): cd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) − 1.

(Lyubeznik, 1993): ´ecd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) + n − 1.

As we learned in the above slide:

ara(X ) ≥ cd(Pⁿ\ X ) + 1.

ara(X ) ≥ ´ecd(Pⁿ\ X ) − n + 1.

QUESTION (Hartshorne, 1970): Which is better???

?

´ecd(Pⁿ\ X ) ≷ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Grothendieck, 1961): cd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) − 1.

(Lyubeznik, 1993): ´ecd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) + n − 1.

As we learned in the above slide:

ara(X ) ≥ cd(Pⁿ\ X ) + 1.

ara(X ) ≥ ´ecd(Pⁿ\ X ) − n + 1.

QUESTION (Hartshorne, 1970): Which is better???

?

´ecd(Pⁿ\ X ) ≷ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Grothendieck, 1961): cd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) − 1.

(Lyubeznik, 1993): ´ecd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) + n − 1.

As we learned in the above slide:

ara(X ) ≥ codim_Pⁿ(X ).

ara(X ) ≥ cd(Pⁿ\ X ) + 1.

ara(X ) ≥ ´ecd(Pⁿ\ X ) − n + 1.

QUESTION (Hartshorne, 1970): Which is better???

?

´ecd(Pⁿ\ X ) ≷ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Grothendieck, 1961): cd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) − 1.

(Lyubeznik, 1993): ´ecd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) + n − 1.

As we learned in the above slide:

ara(X ) ≥ codim_Pⁿ(X ).

ara(X ) ≥ cd(Pⁿ\ X ) + 1.

ara(X ) ≥ ´ecd(Pⁿ\ X ) − n + 1.

QUESTION(Hartshorne, 1970): Which is better???

?

´ecd(Pⁿ\ X ) ≷ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Grothendieck, 1961): cd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) − 1.

(Lyubeznik, 1993): ´ecd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) + n − 1.

As we learned in the above slide:

ara(X ) ≥ codim_Pⁿ(X ).

ara(X ) ≥ cd(Pⁿ\ X ) + 1.

ara(X ) ≥ ´ecd(Pⁿ\ X ) − n + 1.

QUESTION(Hartshorne, 1970): Which is better???

?

´ecd(Pⁿ\ X ) ≷ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Grothendieck, 1961): cd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) − 1.

(Lyubeznik, 1993): ´ecd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) + n − 1.

As we learned in the above slide:

ara(X ) ≥ cd(Pⁿ\ X ) + 1.

ara(X ) ≥ ´ecd(Pⁿ\ X ) − n + 1.

QUESTION(Hartshorne, 1970): Which is better???

?

´ecd(Pⁿ\ X ) ≷ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Grothendieck, 1961): cd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) − 1.

(Lyubeznik, 1993): ´ecd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) + n − 1.

As we learned in the above slide:

ara(X ) ≥ cd(Pⁿ\ X ) + 1.

ara(X ) ≥ ´ecd(Pⁿ\ X ) − n + 1.

QUESTION(Hartshorne, 1970): Which is better???

?

´ecd(Pⁿ\ X ) ≷ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Grothendieck, 1961): cd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) − 1.

(Lyubeznik, 1993): ´ecd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) + n − 1.

As we learned in the above slide:

ara(X ) ≥ cd(Pⁿ\ X ) + 1.

ara(X ) ≥ ´ecd(Pⁿ\ X ) − n + 1.

QUESTION(Hartshorne, 1970): Which is better???

?

´ecd(Pⁿ\ X ) ≷ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Grothendieck, 1961): cd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) − 1.

(Lyubeznik, 1993): ´ecd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) + n − 1.

As we learned in the above slide:

ara(X ) ≥ cd(Pⁿ\ X ) + 1.

ara(X ) ≥ ´ecd(Pⁿ\ X ) − n + 1.

QUESTION(Hartshorne, 1970): Which is better???

?

´ecd(Pⁿ\ X ) ≷ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

S o m e r e s u l t s t o d e a l w i t h t h i s p r o b l e m

(Grothendieck, 1961): cd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) − 1.

(Lyubeznik, 1993): ´ecd(Pⁿ\ X ) ≥ codim_Pⁿ(X ) + n − 1.

As we learned in the above slide:

ara(X ) ≥ cd(Pⁿ\ X ) + 1.

ara(X ) ≥ ´ecd(Pⁿ\ X ) − n + 1.

QUESTION(Hartshorne, 1970): Which is better???

?

´ecd(Pⁿ\ X ) ≷ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

H i s t o r y

(Ogus, 1973): char(k) = 0and X = vd(P^k) ⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

(Newstead, 1980): char(k) > 0 and X = P^s× P^t⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

Since then several authors computed the arithmetical rank of certain varieties (Bruns-Schw¨anzl, Barile, Singh-Walther...) and the ´etale cohomological dimension has always proved at least as effective as the (ordinary) cohomological dimension.

CONJECTURE (Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

H i s t o r y

(Ogus, 1973): char(k) = 0and X = vd(P^k) ⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

(Newstead, 1980): char(k) > 0and X = P^s× P^t⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

Since then several authors computed the arithmetical rank of certain varieties (Bruns-Schw¨anzl, Barile, Singh-Walther...) and the ´etale cohomological dimension has always proved at least as effective as the (ordinary) cohomological dimension.

CONJECTURE (Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

H i s t o r y

(Ogus, 1973): char(k) = 0 andX = vd(P^k) ⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

(Newstead, 1980): char(k) > 0and X = P^s× P^t⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

Since then several authors computed the arithmetical rank of certain varieties (Bruns-Schw¨anzl, Barile, Singh-Walther...) and the ´etale cohomological dimension has always proved at least as effective as the (ordinary) cohomological dimension.

CONJECTURE (Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

H i s t o r y

(Ogus, 1973): char(k) = 0 and X = vd(P^k) ⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

(Newstead, 1980): char(k) > 0and X = P^s× P^t⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

Since then several authors computed the arithmetical rank of certain varieties (Bruns-Schw¨anzl, Barile, Singh-Walther...) and the ´etale cohomological dimension has always proved at least as effective as the (ordinary) cohomological dimension.

CONJECTURE (Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

H i s t o r y

(Ogus, 1973): char(k) = 0 and X = vd(P^k) ⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

(Newstead, 1980): char(k) > 0and X = P^s× P^t⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

Since then several authors computed the arithmetical rank of certain varieties(Bruns-Schw¨anzl, Barile, Singh-Walther...) and the ´etale cohomological dimension has always proved at least as effective as the (ordinary) cohomological dimension.

CONJECTURE (Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

H i s t o r y

(Ogus, 1973): char(k) = 0 and X = vd(P^k) ⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

(Newstead, 1980): char(k) > 0 andX = P^s× P^t⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

Since then several authors computed the arithmetical rank of certain varieties(Bruns-Schw¨anzl, Barile, Singh-Walther...) and the ´etale cohomological dimension has always proved at least as effective as the (ordinary) cohomological dimension.

CONJECTURE (Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

H i s t o r y

(Ogus, 1973): char(k) = 0 and X = vd(P^k) ⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

(Newstead, 1980): char(k) > 0 and X = P^s× P^t⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

Since then several authors computed the arithmetical rank of certain varieties(Bruns-Schw¨anzl, Barile, Singh-Walther...) and the ´etale cohomological dimension has always proved at least as effective as the (ordinary) cohomological dimension.

CONJECTURE (Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

H i s t o r y

(Ogus, 1973): char(k) = 0 and X = vd(P^k) ⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

(Newstead, 1980): char(k) > 0 and X = P^s× P^t⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

Since then several authors computed the arithmetical rank of certain varieties(Bruns-Schw¨anzl, Barile, Singh-Walther...) and the ´etale cohomological dimension has always proved at least as effective as the (ordinary) cohomological dimension.

CONJECTURE(Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

H i s t o r y

(Ogus, 1973): char(k) = 0 and X = vd(P^k) ⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

(Newstead, 1980): char(k) > 0 and X = P^s× P^t⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

Since then several authors computed the arithmetical rank of certain varieties (Bruns-Schw¨anzl, Barile, Singh-Walther...) and the ´etale cohomological dimension has always proved at least as effective as the (ordinary) cohomological dimension.

CONJECTURE(Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

H i s t o r y

(Ogus, 1973): char(k) = 0 and X = vd(P^k) ⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

(Newstead, 1980): char(k) > 0 and X = P^s× P^t⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

Since then several authors computed the arithmetical rank of certain varieties (Bruns-Schw¨anzl, Barile, Singh-Walther...) and the ´etale cohomological dimension has always provedat least as effective asthe (ordinary) cohomological dimension.

CONJECTURE(Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

H i s t o r y

(Ogus, 1973): char(k) = 0 and X = vd(P^k) ⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

(Newstead, 1980): char(k) > 0 and X = P^s× P^t⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

Since then several authors computed the arithmetical rank of certain varieties (Bruns-Schw¨anzl, Barile, Singh-Walther...) and the ´etale cohomological dimension has always proved at least as effective as the (ordinary) cohomological dimension.

CONJECTURE(Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

H i s t o r y

(Ogus, 1973): char(k) = 0 and X = vd(P^k) ⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

(Newstead, 1980): char(k) > 0 and X = P^s× P^t⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

Since then several authors computed the arithmetical rank of certain varieties (Bruns-Schw¨anzl, Barile, Singh-Walther...) and the ´etale cohomological dimension has always proved at least as effective as the (ordinary) cohomological dimension.

CONJECTURE(Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

H i s t o r y

(Ogus, 1973): char(k) = 0 and X = vd(P^k) ⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

(Newstead, 1980): char(k) > 0 and X = P^s× P^t⊆ Pⁿ.

´ecd(Pⁿ\ X ) > cd(Pⁿ\ X ) + n

Since then several authors computed the arithmetical rank of certain varieties (Bruns-Schw¨anzl, Barile, Singh-Walther...) and the ´etale cohomological dimension has always proved at least as effective as the (ordinary) cohomological dimension.

CONJECTURE(Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

T h e r e s u l t

CONJECTURE(Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

(Lyubeznik, 2009): COUNTEREXAMPLE if char(k) > 0

(-, to appear in Trans. Amer. Math. Soc.):

TRUE in characteristic 0, provided that X is smooth

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

T h e r e s u l t

CONJECTURE(Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

(Lyubeznik, 2009): COUNTEREXAMPLE if char(k) > 0

(-, to appear in Trans. Amer. Math. Soc.):

TRUE in characteristic 0, provided that X is smooth

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

T h e r e s u l t

CONJECTURE(Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

(Lyubeznik, 2009): COUNTEREXAMPLE if char(k) > 0

(-, to appear in Trans. Amer. Math. Soc.):

TRUE in characteristic 0, provided that X is smooth

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

T h e r e s u l t

CONJECTURE(Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

(Lyubeznik, 2009): COUNTEREXAMPLE if char(k) > 0

(-, to appear in Trans. Amer. Math. Soc.):

TRUE in characteristic 0, provided that X is smooth

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

T h e r e s u l t

CONJECTURE(Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

(Lyubeznik, 2009): COUNTEREXAMPLE if char(k) > 0

(-, to appear in Trans. Amer. Math. Soc.):

TRUE in characteristic 0, provided that X is smooth

Matteo Varbaro, Universit`a di Genova PhD thesis

ZARISKI TOPOLOGY VS ´ETALE TOPOLOGY

T h e r e s u l t

CONJECTURE(Lyubeznik, 2002): ´ecd(Pⁿ\ X ) ≥ cd(Pⁿ\ X ) + n

(Lyubeznik, 2009): COUNTEREXAMPLE if char(k) > 0

(-, to appear in Trans. Amer. Math. Soc.):

TRUE in characteristic 0, provided that X is smooth

Matteo Varbaro, Universit`a di Genova PhD thesis