Understanding of the defining equations and syzygies via ... · Sijong Kwak (KAIST, Korea)...

Understanding of the defining equations andsyzygies via inner projections and generic initial

ideals

Sijong Kwak (KAIST, Korea)

Syzygies of Algebraic Varieties Workshop for Graduate Students,Postdocs and Young Researchers,

Nov 20-22, 2015, the University of Illinois at Chicago(UIC)

Sijong Kwak (KAIST, Korea) Understanding of the defining equations and syzygies via inner projections and generic initial idealsNovember 21, 2015 1 / 47

X ⊂ P(V ),V ⊂ H0(OX (1)): a nondegenerate, irreducible andreduced variety of dim(X ) = n and codim(X ) = e defined overK = K of char (K ) ≥ 0. If V = H0(OX (1)), one says that X is alinearly normal embedding.

R/IX : the projective coordinate ring of X whereR = K [x0, x1, . . . , xn+e] is a coordinate ring of P(V ) andIX =

⊕m≥0 H0(IX (m)) is the saturated ideal.

There is a basic exact sequence:

0→ R/IX →⊕m≥0

H0(OX (m))→⊕m≥0

H1(IX (m))→ 0

and X is called "projectively normal" if⊕

m≥0 H1(IX (m)) = 0.

0→ R/IX →⊕m≥0

H0(OX (m))→⊕m≥0

H1(IX (m))→ 0

m≥0 H1(IX (m)) = 0.

0→ R/IX →⊕m≥0

H0(OX (m))→⊕m≥0

H1(IX (m))→ 0

m≥0 H1(IX (m)) = 0.

0→ R/IX →⊕m≥0

H0(OX (m))→⊕m≥0

H1(IX (m))→ 0

m≥0 H1(IX (m)) = 0.

0→ R/IX →⊕m≥0

H0(OX (m))→⊕m≥0

H1(IX (m))→ 0

m≥0 H1(IX (m)) = 0.

0→ R/IX →⊕m≥0

H0(OX (m))→⊕m≥0

H1(IX (m))→ 0

m≥0 H1(IX (m)) = 0.

0→ R/IX →⊕m≥0

H0(OX (m))→⊕m≥0

H1(IX (m))→ 0

m≥0 H1(IX (m)) = 0.

Basic Goal: Understand the minimal free resolutions of R/IXand R(X ) =

⊕m≥0 H0(OX (m)) and their associated Betti tables

in terms of geometric invariants.

Defining equations and their relations (called "syzygies") of Xappear in the (unique) minimal free resolution of R/IX :· · · → Li → Li−1 → · · · → L1 → R → R/IX → 0 whereLi =

⊕j R(−i − j)βi,j (X).

Note that βi,j(X ) is the rank of the degree i + j part in Li andβi,j(X ) = dimK TorR

i (R/IX ,K )i+j .

The simplest nontrivial example is a rational normal curveυd (P1) ↪→ Pd . How to compute the minimal free resolution ofυd (P1) ⊂ Pd for d = 3,4 by hand or by using Macaulay 2 ?

⊕j R(−i − j)βi,j (X).

i (R/IX ,K )i+j .

⊕j R(−i − j)βi,j (X).

i (R/IX ,K )i+j .

⊕j R(−i − j)βi,j (X).

i (R/IX ,K )i+j .

⊕j R(−i − j)βi,j (X).

i (R/IX ,K )i+j .

The section ring R(X ) :=⊕

m≥0 H0(OX (m)) has also the following(unique) minimal free resolution as a graded R-module:

· · · → Li → · · · → L1 → L0 →⊕m≥0

H0(OX (m))→ 0

where L0 = R ⊕ R(−1)t ⊕j≥2 R(−j)β0,j .

Note that t = codim(V ,H0(OX (1)) and if X is not linearly normal,the basis elements in H0(OX (1)) \ V should be generators ofR(X ) in degree 1.

In particular, the minimal free resolution of the section ring R(X )also encodes some geometric information on the embeddingX ↪→ P(V ).

· · · → Li → · · · → L1 → L0 →⊕m≥0

H0(OX (m))→ 0

where L0 = R ⊕ R(−1)t ⊕j≥2 R(−j)β0,j .

· · · → Li → · · · → L1 → L0 →⊕m≥0

H0(OX (m))→ 0

where L0 = R ⊕ R(−1)t ⊕j≥2 R(−j)β0,j .

· · · → Li → · · · → L1 → L0 →⊕m≥0

H0(OX (m))→ 0

where L0 = R ⊕ R(−1)t ⊕j≥2 R(−j)β0,j .

Geometric information of the section ring

For example, if the section ring⊕

m≥0 H0(OX (m)) of a smooth varietyX has the following minimal free resolution:

· · · → R(−2)β1,1 → R ⊕ R(−1)t →⊕m≥0

H0(OX (m))→ 0,

i.e. as simple as possible up to the first syzygies,then we have the following properties (due to E. Park-K, 2005):

X is k -normal for all k ≥ t + 1;X is cut out by equations of degree at most t + 2;reg(X ) ≤ max{m + 1, t + 2} where m = reg(OX ).

Note that if t = 0, then X is projectively normal, and so it is trivial.

· · · → R(−2)β1,1 → R ⊕ R(−1)t →⊕m≥0

H0(OX (m))→ 0,

· · · → R(−2)β1,1 → R ⊕ R(−1)t →⊕m≥0

H0(OX (m))→ 0,

· · · → R(−2)β1,1 → R ⊕ R(−1)t →⊕m≥0

H0(OX (m))→ 0,

· · · → R(−2)β1,1 → R ⊕ R(−1)t →⊕m≥0

H0(OX (m))→ 0,

· · · → R(−2)β1,1 → R ⊕ R(−1)t →⊕m≥0

H0(OX (m))→ 0,

· · · → R(−2)β1,1 → R ⊕ R(−1)t →⊕m≥0

H0(OX (m))→ 0,

Let X ⊂ Pn+e be a variety of dim(X ) = n and deg(X ) = d . We havethe Betti table of R/IX associated to the minimal free resolution.

0 1 2 3 · · · i − 1 i i + 1 · · · 40 1 − − − · · · − − − · · · −1 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · β4,1

2 − β1,2 β2,2 β3,2 · · · βi−1,2 βi,2 βi+1,2 · · · β4,2... − − · · · −

. . . · · ·...

... · · ·. . .

j − β1,j β2,j β3,j · · · βi−1,j βi,j βi+1,j · · · β4,j... · · · −

. . . · · ·...

... · · ·. . .

� − β1,� β2,� β3,� · · · βi−1,� βi,� βi+1,� · · · β4,�

4 = the projective dimension of R/IX , 4 ≥ e.� = reg(R/IX ) = reg(X )− 1, where

reg(X ):= min {α | H i(P, IX (α− i)) = 0}.(Eisenbud-Goto Conjecture) � ≤ deg(X )− codim(X ) = d − e.

0 1 2 3 · · · i − 1 i i + 1 · · · 40 1 − − − · · · − − − · · · −1 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · β4,1

2 − β1,2 β2,2 β3,2 · · · βi−1,2 βi,2 βi+1,2 · · · β4,2... − − · · · −

. . . · · ·...

... · · ·. . .

. . . · · ·...

... · · ·. . .

� − β1,� β2,� β3,� · · · βi−1,� βi,� βi+1,� · · · β4,�

0 1 2 3 · · · i − 1 i i + 1 · · · 40 1 − − − · · · − − − · · · −1 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · β4,1

2 − β1,2 β2,2 β3,2 · · · βi−1,2 βi,2 βi+1,2 · · · β4,2... − − · · · −

. . . · · ·...

... · · ·. . .

. . . · · ·...

... · · ·. . .

� − β1,� β2,� β3,� · · · βi−1,� βi,� βi+1,� · · · β4,�

0 1 2 3 · · · i − 1 i i + 1 · · · 40 1 − − − · · · − − − · · · −1 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · β4,1

2 − β1,2 β2,2 β3,2 · · · βi−1,2 βi,2 βi+1,2 · · · β4,2... − − · · · −

. . . · · ·...

... · · ·. . .

. . . · · ·...

... · · ·. . .

� − β1,� β2,� β3,� · · · βi−1,� βi,� βi+1,� · · · β4,�

0 1 2 3 · · · i − 1 i i + 1 · · · 40 1 − − − · · · − − − · · · −1 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · β4,1

2 − β1,2 β2,2 β3,2 · · · βi−1,2 βi,2 βi+1,2 · · · β4,2... − − · · · −

. . . · · ·...

... · · ·. . .

. . . · · ·...

... · · ·. . .

� − β1,� β2,� β3,� · · · βi−1,� βi,� βi+1,� · · · β4,�

By the symmetry of Tor, the graded Betti numbers are also defined viathe Koszul exact sequence of the base field K :V = K 〈x0, · · · , xn+e〉 be the K -vector space in K [x0, . . . , xn+e].Then, TorR

i (R/IX ,K )i+j is the homology of the Koszul complex:

∧i+1V ⊗ (R/IX )j−1ϕi+1,j−1−→ ∧iV ⊗ (R/IX )j

ϕi,j−→ ∧i−1V ⊗ (R/IX )j+1,

where the map is given by ϕi,j(xα1 ∧ xα2 ∧ · · · ∧ xαi ⊗m) =∑1≤µ≤i (−1)µ−1xα1 · · · ∧ ˆxαµ ∧ . . . ∧ xαi ⊗ (xαµ ·m).

Note that the Koszul complex is exact if i > n + e + 1 or j >> 0.

β1,1(X ): the number of quadrics Qi ∈ IX ;β2,1(X ) is the number of linear relations of the form ΣLiQi = 0;β3,1(X ) is the number of linear relations of linear relations;β1,2(X ) is the number of cubic generators of IX .

ϕi,j−→ ∧i−1V ⊗ (R/IX )j+1,

I Many geometric information on X can be read off from the Betti table(e.g. Green conjecture, gonality conjecture, genus, degree etc).

I One can say that X satisfies property N2,p (or Np ) if βi,j(X ) = 0 for1 ≤ i ≤ p, j ≥ 2.

0 1 2 3 · · · p p + 1 · · · 40 1 − − − · · · − · · · −1 − β1,1 β2,1 β3,1 · · · βp,1 βp+1,1 · · · β4,1

2 − − − − · · · − βp+1,2 · · · β4,2

IWe are interested in the Betti numbers in the first linear strand.

0 1 2 3 · · · i − 1 i i + 1 · · · 41 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · β4,1

0 1 2 3 · · · p p + 1 · · · 40 1 − − − · · · − · · · −1 − β1,1 β2,1 β3,1 · · · βp,1 βp+1,1 · · · β4,1

2 − − − − · · · − βp+1,2 · · · β4,2

0 1 2 3 · · · i − 1 i i + 1 · · · 41 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · β4,1

0 1 2 3 · · · p p + 1 · · · 40 1 − − − · · · − · · · −1 − β1,1 β2,1 β3,1 · · · βp,1 βp+1,1 · · · β4,1

2 − − − − · · · − βp+1,2 · · · β4,2

0 1 2 3 · · · i − 1 i i + 1 · · · 41 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · β4,1

0 1 2 3 · · · p p + 1 · · · 40 1 − − − · · · − · · · −1 − β1,1 β2,1 β3,1 · · · βp,1 βp+1,1 · · · β4,1

2 − − − − · · · − βp+1,2 · · · β4,2

0 1 2 3 · · · i − 1 i i + 1 · · · 41 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · β4,1

0 1 2 3 · · · p p + 1 · · · 40 1 − − − · · · − · · · −1 − β1,1 β2,1 β3,1 · · · βp,1 βp+1,1 · · · β4,1

2 − − − − · · · − βp+1,2 · · · β4,2

0 1 2 3 · · · i − 1 i i + 1 · · · 41 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · β4,1

0 1 2 3 · · · p p + 1 · · · 40 1 − − − · · · − · · · −1 − β1,1 β2,1 β3,1 · · · βp,1 βp+1,1 · · · β4,1

2 − − − − · · · − βp+1,2 · · · β4,2

0 1 2 3 · · · i − 1 i i + 1 · · · 41 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · β4,1

Natural Philosophy: More quadrics X has, higher linear syzygies canbe nonzero and only linear syzygies can happen!

Known Facts:

[Green, 1984] If βp,1 6= 0, then h0(IX (2)) ≥(p+1

)= (p+1)p

2 ;[Han-K, 2012] If X satisfies property N2,p, then

h0(IX (2)) ≥(e+1

)−(e+1−p

)= (2e+1−p)p

Recall that the Betti number βi,1 is defined as the homology of

0→ ∧i+1Vϕi+1,0−→ ∧iV ⊗ V

ϕi,1−→ ∧i−1V ⊗ (R/IX )2,

and βi,1 = dimK TorRi (RX ,K )i+1 ≤ dimK (∧iV ⊗ V/ ∧i+1 V ).

Question: What is the sharp bound of βi,1 for a variety X ?

Known Facts:

)= (p+1)p

h0(IX (2)) ≥(e+1

)−(e+1−p

)= (2e+1−p)p

0→ ∧i+1Vϕi+1,0−→ ∧iV ⊗ V

ϕi,1−→ ∧i−1V ⊗ (R/IX )2,

Known Facts:

)= (p+1)p

h0(IX (2)) ≥(e+1

)−(e+1−p

)= (2e+1−p)p

0→ ∧i+1Vϕi+1,0−→ ∧iV ⊗ V

ϕi,1−→ ∧i−1V ⊗ (R/IX )2,

Known Facts:

)= (p+1)p

h0(IX (2)) ≥(e+1

)−(e+1−p

)= (2e+1−p)p

0→ ∧i+1Vϕi+1,0−→ ∧iV ⊗ V

ϕi,1−→ ∧i−1V ⊗ (R/IX )2,

Known Facts:

)= (p+1)p

h0(IX (2)) ≥(e+1

)−(e+1−p

)= (2e+1−p)p

0→ ∧i+1Vϕi+1,0−→ ∧iV ⊗ V

ϕi,1−→ ∧i−1V ⊗ (R/IX )2,

Known Facts:

)= (p+1)p

h0(IX (2)) ≥(e+1

)−(e+1−p

)= (2e+1−p)p

0→ ∧i+1Vϕi+1,0−→ ∧iV ⊗ V

ϕi,1−→ ∧i−1V ⊗ (R/IX )2,

Known Facts:

)= (p+1)p

h0(IX (2)) ≥(e+1

)−(e+1−p

)= (2e+1−p)p

0→ ∧i+1Vϕi+1,0−→ ∧iV ⊗ V

ϕi,1−→ ∧i−1V ⊗ (R/IX )2,

Classical Question: How many possible quadric hypersurfacescontaining X ⊂ P(V ), i.e. ∃ an upper bound of β1,1?

The simplest examples of curves

I A rational normal curve C ⊂ P1+e :0→ H0(IC(2))→ H0(OPe+1(2))→ H0(OC(2))→ 0 and by R-R,h0(IC(2)) = h0(OPe+1(2))−h0(OC(2)) =

)− (2(e +1) +1) =

I An elliptic normal curve C ⊂ P1+e : Similarly, we haveh0(IC(2)) =

)− (2(e + 2) + 1− g(C)) =

)− 1.

� Castelnuovo(1889)h0(IX/Pn+e (2)) ≤

)and “ = ” holds iff X is a variety of minimal

degree, i.e. deg(X ) = e + 1.

)− (2(e +1) +1) =

)− (2(e + 2) + 1− g(C)) =

)− 1.

)− (2(e +1) +1) =

)− (2(e + 2) + 1− g(C)) =

)− 1.

)− (2(e +1) +1) =

)− (2(e + 2) + 1− g(C)) =

)− 1.

)− (2(e +1) +1) =

)− (2(e + 2) + 1− g(C)) =

)− 1.

)− (2(e +1) +1) =

)− (2(e + 2) + 1− g(C)) =

)− 1.

Upper bound of the number of quadrics

Castelnuovo’s simple proof.

Γ = X ∩ Pe is a set of d-points in general position for general Pe.Since d ≥ e + 1, take a subset Γ′ = {p1,p1, . . . ,pe+1} ⊂ Γ ⊂ Pe.h0(IX (2)) ≤ h0(IΓ(2)) ≤ h0(IΓ′(2)) =

)− (e + 1) =

� Further results in this method.

(Fano, 1894) Unless X is VMD,h0(IX (2)) ≤

)− 1 and “ = ” holds iff X is a del Pezzo variety

(i.e. arithmetically Cohen-Macaulay and deg(X ) = e + 2).Note that the curve sections of a VMD and a del Pezzo variety arethe rational normal curve, smooth elliptic normal curve and arational nodal curve.

)− (e + 1) =

Our observation using inner projections.

Consider πq : X 99K Xq ⊂ Pn+e−1 from a smooth point q ∈ X :

Blq(X ) ' X̃π̃q

��X ⊂ Pn+e // Xq = πq(X \ {q}) ⊂ Pn+e−1.

LemmaX n ⊂ Pn+e: irreducible and reduced (not necessarilysmooth). Then, we have the following:

1 h0(IXq (2)) ≤ h0(Pn+e, IX (2)) ≤ h0(IXq (2)) + e .2 In the second inequality, ” = ” holds iff X is a local complete

intersection of quadrics.

Blq(X ) ' X̃π̃q

��X ⊂ Pn+e // Xq = πq(X \ {q}) ⊂ Pn+e−1.

Blq(X ) ' X̃π̃q

��X ⊂ Pn+e // Xq = πq(X \ {q}) ⊂ Pn+e−1.

Blq(X ) ' X̃π̃q

��X ⊂ Pn+e // Xq = πq(X \ {q}) ⊂ Pn+e−1.

Blq(X ) ' X̃π̃q

��X ⊂ Pn+e // Xq = πq(X \ {q}) ⊂ Pn+e−1.

Blq(X ) ' X̃π̃q

��X ⊂ Pn+e // Xq = πq(X \ {q}) ⊂ Pn+e−1.

[simple proof] Note that h0(IXq (2)) ≤ h0(Pn+e, IX (2)) is trivial.For the second inequality, we assume a smooth pointq = (1,0, · · · ,0) ∈ X .A quadratic form vanishing on X has no x2

0 -term and can be written bythe (Gauss) elimination as follows:

x0L1 + Q1, · · · , x0Lt + Qt , Q̃1, · · · , Q̃µ

where Li is a linear form and Qi , Q̃j are quadrics in k [x1, x2, · · · , xn+e].Note that t ≤ e and µ = h0(IXq (2)). Therefore, by successive innerprojections from smooth points up to a hypersurface Z in Pn+1,

h0(IX (2)) ≤ h0(IXq (2)) + e ≤ 1 + 2 + · · ·+ e =

(e + 1

Furthermore, h0(IX (2)) ≤(e+1

)iff a hypersurface Z in Pn+1 is quadric.

x0L1 + Q1, · · · , x0Lt + Qt , Q̃1, · · · , Q̃µ

h0(IX (2)) ≤ h0(IXq (2)) + e ≤ 1 + 2 + · · ·+ e =

(e + 1

x0L1 + Q1, · · · , x0Lt + Qt , Q̃1, · · · , Q̃µ

h0(IX (2)) ≤ h0(IXq (2)) + e ≤ 1 + 2 + · · ·+ e =

(e + 1

x0L1 + Q1, · · · , x0Lt + Qt , Q̃1, · · · , Q̃µ

h0(IX (2)) ≤ h0(IXq (2)) + e ≤ 1 + 2 + · · ·+ e =

(e + 1

x0L1 + Q1, · · · , x0Lt + Qt , Q̃1, · · · , Q̃µ

h0(IX (2)) ≤ h0(IXq (2)) + e ≤ 1 + 2 + · · ·+ e =

(e + 1

More general inequality (Han-K, 2015)

βi,1(X ) ≤ βi,1(Xq) + βi−1,1(Xq) +(e

), i ≥ 1.

The equality holds for i ≤ p if X satisfies property N2, p.

Using the above inequality, we have the following:Theorem X ⊂ Pn+e: irreducible and reduced (not necessarilysmooth).

βp,1(X ) ≤ p(

e + 1p + 1

), p ≥ 1

and the following are equivalent:(a) X is a variety of minimal degree;(b) one βp,1(X ) achieves the maximum for some p ≥ 1;(c) X is 2-regular ACM.

βi,1(X ) ≤ βi,1(Xq) + βi−1,1(Xq) +(e

), i ≥ 1.

βp,1(X ) ≤ p(

e + 1p + 1

), p ≥ 1

βi,1(X ) ≤ βi,1(Xq) + βi−1,1(Xq) +(e

), i ≥ 1.

βp,1(X ) ≤ p(

e + 1p + 1

), p ≥ 1

βi,1(X ) ≤ βi,1(Xq) + βi−1,1(Xq) +(e

), i ≥ 1.

βp,1(X ) ≤ p(

e + 1p + 1

), p ≥ 1

βi,1(X ) ≤ βi,1(Xq) + βi−1,1(Xq) +(e

), i ≥ 1.

βp,1(X ) ≤ p(

e + 1p + 1

), p ≥ 1

We also characterize Fano varieties as follows:Theorem [Han-K, 2015]Unless X is a variety of minimal degree, then we have

βp,1(X ) ≤ p(

e + 1p + 1

ep − 1

)for all p ≤ e ,

and the following are equivalent:(a) X is a del Pezzo variety;(c) one βp,1(X ) achieves the maximum for some 1 ≤ p ≤ e − 1;(d) all the βp,1(X ) for 1 ≤ p ≤ e − 1 achieve the maximum;

βp,1(X ) ≤ p(

e + 1p + 1

ep − 1

)for all p ≤ e ,

βp,1(X ) ≤ p(

e + 1p + 1

ep − 1

)for all p ≤ e ,

Elimination mapping cone sequence and the partial eliminationideal theory due to M. Green are very useful to prove the abovefundamental inequality on the Betti numbers βi,1(X ) and βi,1(Xq).

I Elimination mapping cone sequenceLet S = k [x1, . . . , xn+e] ⊂ R = k [x0, x1 . . . , xn+e]Let M be a graded R-module (so, M is also a graded S-module).Then, we have a natural long exact sequence:TorR

i (M)i+j → TorSi−1(M)i−1+j

×x0→ TorSi−1(M)i−1+j+1 → TorR

i−1(M)i−1+j+1whose connecting homomorphism is induced by the multiplication map×x0 : M(−1)→ M.

This long exact sequence is useful to study the syzygies ofprojections.We can prove that if X satisfies property Np, then there is nop + 2-secant p-plane to X . (A proof is on the blackboard!)

Varieties of minimal degree

X n ⊂ Pn+e: a variety (not necessarily smooth) of degree d .Note that d ≥ e + 1. X is called "minimal degree variety" if d = e + 1.I The simplest Betti table with βi,1 = i ·

(e+1i+1

0 1 2 3 · · · i − 1 i i + 1 · · · e0 1 − − − · · · − − − · · · −1 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · βe,1

Table 1 minimal degree varieties

I A VMD has a rational normal curve section and they have the sameBetti table.

(e+1i+1

On the other hand, P. del Pezzo(1886) and E. Bertini(1907) classifiedall varieties of minimal degree:I X is of minimal degree iff X is (a cone of) one of the following;(a) a quadric hypersurface;(b) a Veronese surface ν2(P2) in P5;(c) a rational normal scroll, i.e. P(E) ↪→ PΣai +d , whereE '

⊕di=0OP1(ai),ai ≥ 1.

Also See the paper "On Varieties of Minimal Degree (A centennialAccount)-1987" due to D. Eisenbud and J. Harris.

del Pezzo varieties

X is called a del Pezzo variety if d = e + 2 and depth(X ) = n + 1.I The (next-to-simplest) Betti table of a del Pezzo variety:

0 1 2 3 · · · i · · · e − 1 e0 1 − − − · · · − · · · − −1 − β1,1 β2,1 β3,1 · · · βi,1 · · · βe−1,1 −2 − − − − · · · − − − βe,2 = 1

Table 2 del Pezzo variety with βi,1(X ) = i(e+1

)−( e

I A del Pezzo variety has an ellitic normal curve section or a rationalnodal curve section and they have the same Betti table.

del Pezzo varieties

0 1 2 3 · · · i · · · e − 1 e0 1 − − − · · · − · · · − −1 − β1,1 β2,1 β3,1 · · · βi,1 · · · βe−1,1 −2 − − − − · · · − − − βe,2 = 1

)−( e

del Pezzo varieties

0 1 2 3 · · · i · · · e − 1 e0 1 − − − · · · − · · · − −1 − β1,1 β2,1 β3,1 · · · βi,1 · · · βe−1,1 −2 − − − − · · · − − − βe,2 = 1

)−( e

del Pezzo varieties

T. Fujita classified smooth del Pezzo varieties into 8 types.T. Fujita also showed that (singular) normal del Pezzo cases aredivisors of some specific type in rational normal scrolls.(M. Brodmann-P. Schenzel) Every non-normal del Pezzo variety Xcomes from outer projection of a minimal degree variety X̃ from apoint q in Sec(X̃ ) \ X̃ satisfying dim Σq(X̃ ) = dim X̃ − 1 where thesecant locus

Σq(X̃ ) := {x ∈ X̃ | π−1q (πq(x)) has length at least 2 }.

Thus, a non-normal del Pezzo variety has a rational nodal curvesection.

del Pezzo varieties

Classification of non-normal del Pezzo varieties

M. Brodmann and E. Park showed that ∃ only 8 types ofnon-normal del Pezzo which are not cones:

(i) Outer projections of a rational normal curve C ⊂ Pa (a > 2) fromq ∈ Sec(C),

(ii) Outer projections of the Veronese surface υ2(P2) ⊂ P5 fromq ∈ Sec(υ2(P2)),

(iii) Outer projections of a smooth cubic surface scroll S(1,2) in P4,(iv) Outer projections of a smooth rational normal scroll S(1,b) in

Pb+2 (b > 2) from q ∈ Join(S(1),S(1,b)),(v) Outer projections of a smooth quartic surface scroll S(2,2) in P5

from q ∈ P2 × P1,(vi) Outer projections of a smooth surface scroll S(2,b) in Pb+3 (b > 2)

from q ∈ 〈S(2)〉,(vii) Outer projections of a smooth 3-fold scroll S(1,1,1) in P5,(viii) Outer projections of a smooth 3-fold scroll S(1,1, c) in Pc+4 with

c > 1 from q ∈ 〈S(1,1)〉.

Asymptotic behavior of βp,1 for smooth curves

Let C be a smooth curve of genus g and the gonality g0.Suppose deg(L) ≥ 4g − 3 and r = h0(C,L)− 1.

(a) βp,1(C) 6= 0⇐⇒ 1 ≤ p ≤ r − g0 (Ein-Lazarsfeld);(b) βp,2(C) 6= 0⇐⇒ r − g ≤ p ≤ r − 1 (Green and Schreyer).

0 1 · · · r − g − 1 r − g · · · r − g0 · · · r − 10 1 − · · · − · · · − − − −1 − β1,1 · · · βr−g−1,1 βr−g,1 · · · βr−g0,1 − −2 − − · · · − βr−g,2 · · · · · · βr−1,2

IWe can also get the upper bound of βp,1 by using inner projectionmethod as follows:

βp,1(C) ≤ p(

e + 1p + 1

(e + 1p + 1

e + 2− g0

)− (g0 − 1)

(e + 1

βp,1(C) ≤ p(

e + 1p + 1

(e + 1p + 1

e + 2− g0

)− (g0 − 1)

(e + 1

βp,1(C) ≤ p(

e + 1p + 1

(e + 1p + 1

e + 2− g0

)− (g0 − 1)

(e + 1

βp,1(C) ≤ p(

e + 1p + 1

(e + 1p + 1

e + 2− g0

)− (g0 − 1)

(e + 1

βp,1(C) ≤ p(

e + 1p + 1

(e + 1p + 1

e + 2− g0

)− (g0 − 1)

(e + 1

The graded Betti table of R/IX

0 1 2 3 · · · i − 1 i i + 1 · · · 40 1 − − − · · · − − − · · · −1 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · β4,1

2 − β1,2 β2,2 β3,2 · · · βi−1,2 βi,2 βi+1,2 · · · β4,2... − − · · · −

. . . · · ·...

... · · ·. . .

. . . · · ·...

... · · ·. . .

� − β1,� β2,� β3,� · · · βi−1,� βi,� βi+1,� · · · β4,�

In the first linear strand, we summarize the following facts:

βp,1 = 0 for p > e;The maximal upper bounds for a VMD;The next to maximal upper bounds for a del Pezzo.

0 1 2 3 · · · i − 1 i i + 1 · · · 40 1 − − − · · · − − − · · · −1 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · β4,1

2 − β1,2 β2,2 β3,2 · · · βi−1,2 βi,2 βi+1,2 · · · β4,2... − − · · · −

. . . · · ·...

... · · ·. . .

. . . · · ·...

... · · ·. . .

� − β1,� β2,� β3,� · · · βi−1,� βi,� βi+1,� · · · β4,�

0 1 2 3 · · · i − 1 i i + 1 · · · 40 1 − − − · · · − − − · · · −1 − β1,1 β2,1 β3,1 · · · βi−1,1 βi,1 βi+1,1 · · · β4,1

2 − β1,2 β2,2 β3,2 · · · βi−1,2 βi,2 βi+1,2 · · · β4,2... − − · · · −

. . . · · ·...

... · · ·. . .

. . . · · ·...

... · · ·. . .

� − β1,� β2,� β3,� · · · βi−1,� βi,� βi+1,� · · · β4,�

The structure of the Betti table II (Regularity)

Let X n ⊂ Pn+e be a non-degenerate projective variety of dim n,codim e, and degree d , and H be a general hyperplane section.

Definition

X is called m-regular if the following two conditions hold:1 H0(OPn+e (m − 1))� H0(OX (m − 1)) is surjective, i.e.

X is (m − 1)-normal;2 H i (OX (m − 1− i)) = 0 for all i ≥ 1, i.e.OX is (m − 1)-regular with respect to OX (1).

reg(X ) (Castelnuovo-Mumford regularity of X ) is the smallestnumber m such that X is m-regular;

Geometric Regularity Bound

reg(X ) ≤ d − e + 1 (Eisenbud-Goto conjecture).

Definition

Proposition (Birational double point formula)Let ϕ : V n → Mn+1 be a morphism of smooth projective varieties suchthat ϕ : V →W := ϕ(V ) ⊂ M is birational. Then,

ϕ∗(KM + W )− KV ∼ D − E ,

where D and E are effective divisors on V such that E is ϕ-exceptional.Moreover, if ϕ is isomorphic at x ∈ V , then x /∈ Supp(D − E).

Proof. see Lemma 10.2.8. in Positivity in Algebraic Geometry II.

IWe apply this formula to a general projection of smooth varieties:

πΛ : X � XΛ ⊂ Pn+1,Λ = Pe−2 and Λ ∩ X = ∅.

Note that deg(X ) = deg(XΛ) = d .

ϕ∗(KM + W )− KV ∼ D − E ,

Theorem [Noma, Park and K-]Let X be smooth and L is very ample. We have the following:

H i(X ,L⊗(d−e−i)) = H i(OX (d − e − i)) = 0, i ≥ 1

where d = deg(X ) and e = codim(X ) in the embedding ofX ⊂ P(H0(L)).

Corollary

regR(X ,L) ≤ d − e, i.e. βp,q(R(X )) = 0,∀q ≥ d − e + 1.

Corollary

Duality Theorem for syzygies

Consider the following diagram

DΛ ⊂ X

��

� � // Pn+e

��ZΛ ⊂ XΛ

� � // Pn+1.

Since πΛ is finite, so πΛ-exceptional divisor E = ∅ and thenon-isomorphic locus DΛ (as a divisor) is linearly equivalent to

B2 := (d − n − 2)H − KX ,

which is called a double point divisor arising from πΛ : X � XΛ ⊂ Pn+1.Let V ⊂ H0(OX (B2)) be a subspace spanned by geometric sections(i.e. for s ∈ V , div(s) = DΛ coming from an actual outer projection).

DΛ ⊂ X

��

� � // Pn+e

��ZΛ ⊂ XΛ

� � // Pn+1.

B2 := (d − n − 2)H − KX ,

DΛ ⊂ X

��

� � // Pn+e

��ZΛ ⊂ XΛ

� � // Pn+1.

B2 := (d − n − 2)H − KX ,

DΛ ⊂ X

��

� � // Pn+e

��ZΛ ⊂ XΛ

� � // Pn+1.

B2 := (d − n − 2)H − KX ,

DΛ ⊂ X

��

� � // Pn+e

��ZΛ ⊂ XΛ

� � // Pn+1.

B2 := (d − n − 2)H − KX ,

Proposition(Mumford)V is a basepoint-free subsystem in H0(OX (B2)). In particular,

H i(OX (d − i − 1)) = 0, i ≥ 1.

Proof. For x ∈ X , take a general linear space Λx such thatΛx ∩ (Tx (X )n ∪ Cone(x ,X )n+1) = ∅.Then πΛx : X � XΛx ⊂ Pn+1 isisomorphic near x and x /∈ DΛx and B2 := (d − n − 2)H − KX isbasepoint-free. Since (d − 1− i)H = KX + (n + 1− i)H + B2, byKodaira vanishing theorem H i(OX (d − i − 1)) = 0, i ≥ 1. �

RemarkregH(OX ) ≤ d − 1 for a smooth projective variety X .

H i(OX (d − i − 1)) = 0, i ≥ 1.

RemarkB.Ilic(1995) proved that B2 is big unless X is the Segre embedding ofP1 × Pn−1.By Kawamata-Viehweg vanishing, we have

H i(OX (d − 2− i)) = 0, i ≥ 1.

Double point divisors from inner projectionsLet x1, . . . , xe−1 ∈ X be general points, and let Λ := 〈x1, . . . , xe−1〉.Consider the inner projection at Λ and the blow-up at x1, . . . , xe−1.

X̃ σ //

��

� � // Pn+e

��X Λ� � // Pn+1

Note that deg(X Λ) = deg(X )− (e − 1) = d − (e − 1).

H i(OX (d − 2− i)) = 0, i ≥ 1.

X̃ σ //

��

� � // Pn+e

��X Λ� � // Pn+1

H i(OX (d − 2− i)) = 0, i ≥ 1.

X̃ σ //

��

� � // Pn+e

��X Λ� � // Pn+1

H i(OX (d − 2− i)) = 0, i ≥ 1.

X̃ σ //

��

� � // Pn+e

��X Λ� � // Pn+1

H i(OX (d − 2− i)) = 0, i ≥ 1.

X̃ σ //

��

� � // Pn+e

��X Λ� � // Pn+1

Assume that X is neither a scroll over a curve nor a second Veronesesurface. Then π̃ : X̃ � X Λ has no exceptional divisor(A. Noma,2013).So, by the birational double point formula, we obtain an effectivedivisor D(π̃) which is the non-isomorphic locus of π̃.Then,

D(π) := σ(D(π̃)|X̃\E1∪···∪Ee−1

is an effective divisor linearly equivalent to

B2,inn := (d − n − e − 1)H − KX ,

which is called a double point divisor from inner projection.

D(π) := σ(D(π̃)|X̃\E1∪···∪Ee−1

B2,inn := (d − n − e − 1)H − KX ,

D(π) := σ(D(π̃)|X̃\E1∪···∪Ee−1

B2,inn := (d − n − e − 1)H − KX ,

D(π) := σ(D(π̃)|X̃\E1∪···∪Ee−1

B2,inn := (d − n − e − 1)H − KX ,

If X is neither a scroll over a curve, a second Veronese surface, nor aRoth variety, then, C(X ) is finite and we have the following:

Proposition (Noma)B2,inn is semiample, i.e. some power of B2,inn is basepoint-free.

Proof: For any point x ∈ X \ C(X ), by varying centers, we can take aninner projection π : X 99K X Λ ⊂ Pn+1 isomorphic at x . �

Note that b.p.f. ⇒ semiample⇒ nef.

Corollary

regH(OX ) ≤ d − e if X is neither a scroll over a curve, a secondVeronese surface, nor a Roth variety.

Proof: (d − e − i)H = KX + (n + 1− i)H + B2,inn.

Corollary

PropositionWe have the following OX -regularity bound for smooth varieties;

X = υ2(P2) ⊂ P4 or in P5 ⇒ reg(OX ) = 1;X : Roth variety⇒ reg(OX ) ≤ d − e;X : a Scroll over a curve of genus g:

g = 0⇒ reg(OX ) = 1;g = 1⇒ reg(OX ) = 2;g ≥ 2⇒ reg(OX ) ≥ d − e − 2.

Therefore, the second part of Eisenbud-Goto conjecture(i.e.reg(OX ) ≤ d − e is proved for smooth varieties. Thus,

reg(R(X )) ≤ d − e.

But, this part looks to be intrinsic without regard to the projectiveembedding of X .

reg(R(X )) ≤ d − e.

Generalization to ND(1)-subscheme

I Our results can be generalized to ND(1)-subschemes using genericinitial ideals in graded reverse lexicographic order.

DefinitionA closed subscheme X n ⊂ Pn+e is called ND(1)-subscheme if X ∩ Λ isnondegenerate for a general Λ of dimension e ≤ dim Λ ≤ n + e.

Ia ND(1)-subscheme is not necessarily to be irreducible, reduced orequi-dimensional in general.

Example (ND(1)-subschemes)

Nondegenerate varieties;Connected in codimension 1 algebraic sets, i.e. X isequidimensional and each components are ordered such thatXi ∩ Xi+1 is of codim 1 in X ;Algebraic sets whose one component is nondegenerate;

Non–ND(1) subschemesISkew lines in P3 is non-ND(1) and its Betti table is as follows:

0 1 2 30 1 − − −1 − 4 4 1

This is a counter-example for all the previous results.I Two planes meeting at one point in P4 is also non-ND(1).I For a ND(1) subscheme in Pn+e, it is easy to show deg(X ) ≥ e + 1.So, we can define a minimal degree ND(1)-scheme.I Take a diagram explaining the category of ND(1).

0 1 2 30 1 − − −1 − 4 4 1

Theorem [Ahn, Han and K-, preprint]X n ⊂ Pn+e: a ND(1) subscheme, defined over K = K of char (K ) = 0.Then, we have the following upper bound:

βp,1(X ) ≤ p(

e + 1p + 1

), p ≥ 1

and the following are equivalent:(a) X is a ND(1) subscheme of minimal degree;(b) h0(IX/Pn+e (2)) =

(c) one βp,1(X ) achieves the maximum for some p ≥ 1;(d) X has ACM 2-linear resolution.

Corollary βp,1(X ) = 0, p ≥ e + 1 for any ND(1) subscheme X .

βp,1(X ) ≤ p(

e + 1p + 1

), p ≥ 1

βp,1(X ) ≤ p(

e + 1p + 1

), p ≥ 1

βp,1(X ) ≤ p(

e + 1p + 1

), p ≥ 1

βp,1(X ) ≤ p(

e + 1p + 1

), p ≥ 1

Ideas of a proof in case char(K ) = 0

Generic initial idealFor g = (gij) ∈ GLn+1(K ), consider g(I) = {g · f | f ∈ I} whereg · f = f (gx0,gx1, · · · ,gxn+e) and gxi =

∑0≤k≤n+e gikxk .

Then, due to Galligo and Bayer-Stillman, inτ (g(I)) is constant for ageneral change g. We will call this the generic initial ideal of I w.r.t τ .

I Gin(IX ): the generic initial ideal of IX in reverse lexicographic order.Then, the Hilberet functions of R/IX and R/Gin(IX ) are the same.

Gin(IX ) is Borel-fixed, i.e. if m ∈ Gin(IX ) is divisible by xj , thenxixj

m ∈ Gin(IX ) for i < j ;

βi,j(R/IX ) ≤ βi,j(R/Gin(IX )) (Cancellation principle).reg(IX ) = reg(Gin(IX )) =the maximal degree of generators ofGin(IX );

I Gin(IX∩H) = (Gin(IX )| xn+e→0)|xn+e−1→1 for a general hyperplane H(Bayer-Stillman or M. Green);(This is the main reason we use the reverse lexicographic order).

I For a generic initial ideal Gin(IX ), we can compute the graded Bettinumber βi,j(R/Gin(IX )) by the combinatorial method due to theEliahou-Kervaire Theorem.

βi,j(R/Gin(IX )) =∑

T∈G(Gin(IX ))j+1

(max Ti−1

)(Eliahou-Kervaire), where

I max T := max{t | xt divides T};I G(Gin(IX ))m: the set of minimal generators of degree m.

Recall again thatβi,j(R/IX ) ≤ βi,j(R/Gin(IX )).

T∈G(Gin(IX ))j+1

(max Ti−1

T∈G(Gin(IX ))j+1

(max Ti−1

T∈G(Gin(IX ))j+1

(max Ti−1

T∈G(Gin(IX ))j+1

(max Ti−1

I [Important fact from ND(1)-property]

xixj 6∈ Gin(IX ),0 ≤ i ≤ e − 1,e ≤ j ≤ n + e

by the non-degenerate condition and Bayer-Stillman theoremGin(IX∩H) = (Gin(IX )| xn+e→0)|xn+e−1→1. Therefore,

By Eliahou-Kervaire, βi,1(R/Gin(IX )) = 0 for i > e.Similarly, Gin(IX ) ⊂ (x0, x1, . . . , xe−1)2 andβi,1(R/IX ) ≤ βi,1(R/Gin(IX )) ≤ βi,1(R/(x0, . . . , xe−1)2) = i

(e+1i+1

Note that the equality holds for some i ≥ 1 if and only ifGin(IX ) = (x0, x1, . . . , xe−1)2 if and only if deg(X ) = e + 1.

(e+1i+1

Our problems on syzygies on cubic generators

Let X n ⊂ Pn+e be any variety over K . Suppose that (IX )2 = 0. Then,(a) What can we say about upper bounds for βp,2(X ) for any p ≥ 2

and Kp,2 Theorem (i.e., βp,2(X ) = 0 for p > e under someconditions.

(b) Classify or characterize (near) boundary cases for the upperbounds of βp,2(X ).

DefinitionA closed subscheme X n ⊂ Pn+e is called ND(2)-subscheme if for ageneral Λ of dimension e ≤ dim Λ ≤ n + e, X ∩ Λ is not contained in aquadric.

Nondegenerate linearly normal curve;Algebraic sets whose one component is a ND(2)-subscheme.

Consider the following Betti table starting from cubic generators for aND(2)-subscheme X :

0 1 2 3 · · · i − 1 i i + 1 · · · 40 1 − − − · · · − − − · · · −1 − − − − · · · − − − · · · −2 − β1,2 β2,2 β3,2 · · · βi−1,2 βi,2 βi+1,2 · · · β4,2... − − · · · −

. . . · · ·...

... · · ·. . .

j − β1,j β2,j β3,j · · · βi−1,j βi,j βi+1,j · · · β4,j

I It is easy to show deg(X ) ≥(e+2

)for a ND(2)-scheme X in Pn+e,.

0 1 2 3 · · · i − 1 i i + 1 · · · 40 1 − − − · · · − − − · · · −1 − − − − · · · − − − · · · −2 − β1,2 β2,2 β3,2 · · · βi−1,2 βi,2 βi+1,2 · · · β4,2... − − · · · −

. . . · · ·...

... · · ·. . .

0 1 2 3 · · · i − 1 i i + 1 · · · 40 1 − − − · · · − − − · · · −1 − − − − · · · − − − · · · −2 − β1,2 β2,2 β3,2 · · · βi−1,2 βi,2 βi+1,2 · · · β4,2... − − · · · −

. . . · · ·...

... · · ·. . .

Upper bound of βp,2(X ) and Kp,2 Theorem

Theorem (Ahn, Han and K-, preprint)

Suppose that X n ⊂ Pn+e is a ND(2) subscheme, defined over K = Kof char (K ) = 0. Then,(e+2

)≤ deg(X ) and h0(IX (3)) ≤

In general, βp,2(X ) ≤(p+1

)(e+2p+2

)for p ≥ 1.

For the extremal cases, the following are equivalent:(a) deg(X ) =

(b) h0(IX (3)) =(e+2

(c) one of βp,2(X ) attains “=” for 1 ≤ p ≤ e ;(d) IX has ACM 3-linear resolution.

This also gives a natural Kp,2 theorem generalizing Kp,1-theorembecause βp,2(X ) = 0 for p > e.

)≤ deg(X ) and h0(IX (3)) ≤

)(e+2p+2

)for p ≥ 1.

(b) h0(IX (3)) =(e+2

)≤ deg(X ) and h0(IX (3)) ≤

)(e+2p+2

)for p ≥ 1.

(b) h0(IX (3)) =(e+2

ACM varieties with 2-linear or 3 linear resolutions

For ND(1)-varieties, deg(X ) = e + 1⇔ X is 2-linear ACM.This is a minimal degree variety in a category of ND(1)-varieties.

0 1 2 3 · · · i · · · e − 1 e0 1 − − − · · · − · · · − −1 − β1,1 β2,1 β3,1 · · · βi,1 · · · βe−1,1 βe,1

For ND(2)-varieties, deg(X ) =(e+2

)⇔ X is 3-linear ACM.

This has a minimal degree in a category of ND(2)-varieties.

0 1 2 3 · · · i · · · e − 1 e0 1 − − − · · · − · · · − −1 − − − − · · · − · · · − −2 − β1,2 β2,2 β3,2 · · · βi,2 · · · βe−1,2 βe,2

0 1 2 3 · · · i · · · e − 1 e0 1 − − − · · · − · · · − −1 − β1,1 β2,1 β3,1 · · · βi,1 · · · βe−1,1 βe,1

0 1 2 3 · · · i · · · e − 1 e0 1 − − − · · · − · · · − −1 − − − − · · · − · · · − −2 − β1,2 β2,2 β3,2 · · · βi,2 · · · βe−1,2 βe,2

0 1 2 3 · · · i · · · e − 1 e0 1 − − − · · · − · · · − −1 − β1,1 β2,1 β3,1 · · · βi,1 · · · βe−1,1 βe,1

0 1 2 3 · · · i · · · e − 1 e0 1 − − − · · · − · · · − −1 − − − − · · · − · · · − −2 − β1,2 β2,2 β3,2 · · · βi,2 · · · βe−1,2 βe,2

I Main question at this point is to give geometric classification/orcharacterization of ACM varieties with 3-linear resolution?

[Examples of varieties having ACM 3-linear resolution]

(a) Hypercubic (e = 1);(b) 3-minors of 4× 4 generic symmetric matrix (i.e. Sec(v2(P3)) ⊂ P9);(c) 3-minors of 3× (e + 2) sufficiently generic matrices (e.g.

Sec(RNS)).

ND(k) subschemes

In general, we can define a ND(k) subscheme X in Pn+e whose Bettitable is the following: (the k -th strand is the first nonzero strand!)

0 1 2 3 · · · i i + 1 · · · 40 1 − − − · · · − − · · · −1 − − − − · · · − − · · · −... − − · · · −

. . . · · ·...

... · · ·k − β1,k β2,k β3,k · · · βi,k βi+1,k · · · β4,k

k + 1 − β1,k+1 β2,k+1 β3,k+1 · · · βi,k+1 βi+1,k+1 · · · β4,k+1... · · · −

. . . · · ·... · · · · · ·

Example (ND(k)-subschemes)

Algebraic sets whose one component is a ND(k)-scheme;(k + 1)-minors of (k + 1)× (e + k) sufficiently generic matrices(e.g. Seck (RNS)).

ND(k) subschemes

In general, we can define a ND(k) subscheme X in Pn+e whose Bettitable is the following: (the k -th strand is the first nonzero strand!)

0 1 2 3 · · · i i + 1 · · · 40 1 − − − · · · − − · · · −1 − − − − · · · − − · · · −... − − · · · −

. . . · · ·...

... · · ·k − β1,k β2,k β3,k · · · βi,k βi+1,k · · · β4,k

k + 1 − β1,k+1 β2,k+1 β3,k+1 · · · βi,k+1 βi+1,k+1 · · · β4,k+1... · · · −

. . . · · ·... · · · · · ·

Example (ND(k)-subschemes)

Algebraic sets whose one component is a ND(k)-scheme;(k + 1)-minors of (k + 1)× (e + k) sufficiently generic matrices(e.g. Seck (RNS)).

Similarly, we have a following theorem:

Theorem (Ahn, Han and K-)

Suppose that X n ⊂ Pn+e is a ND(k) subscheme, defined over K = Kof char (K ) = 0. Then,(e+k

)≤ deg(X ) and h0(IX (k)) ≤

(e+kk+1

In general, βp,k (X ) ≤(p+k−1

)(e+kp+k

)for p ≥ 1.

(b) h0(IX (k + 1)) =(e+k

(c) one of βp,k (X ) attains “=” for 1 ≤ p ≤ e ;(d) IX has ACM (k + 1)-linear resolution.

I There is a filtration of categories of ND(k)-schemes:

· · ·ND(k) schemes ⊂ · · · ⊂ ND(2) schemes ⊂ ND(1) schemes .

(e+kk+1

)(e+kp+k

)for p ≥ 1.

(b) h0(IX (k + 1)) =(e+k

(e+kk+1

)(e+kp+k

)for p ≥ 1.

(b) h0(IX (k + 1)) =(e+k

Classification

I In each category of ND(k) subschemes, the minimal degree is(e+kk

)and the minimal regularity is k + 1.

I Only ACM varieties with (k + 1)-linear resolution have minimaldegree and minimal regularity.

I Classification of such varieties is very far from being complete.

Classification

Understanding of the defining equations and syzygies via ... · Sijong Kwak (KAIST, Korea)...

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