ABSTRACT : PROJECTIONS AND LINEAR SYZYGIESYOUNGOOK CHOI (SEOUL NATIONAL UNIVERSITY)

Let X ⊂ P(H 0 (L)) be a smooth projective variety embedded by thecomplete linear system of a very ample line bundle L on X. One can ask
detailed information about defining equations of X, i.e. their syzygies of X.
As M. Green defined in [4], we can say that L satisfies property N0 if it gives
the projectively normal embedding and L satisfies property N1 if property
N0 holds and X is cut out by quadrics. In general, L satisfies property
Np , p ≥ 1 if X is projectively normal and the projective coordinate ringS(X) of X has the following minimal free resolution of the simplest type asa graded S-module;→ · · · → Ep → Ep−1 → · · · → E1 → S → S(X) → 0L
where S = Sym(H 0 (L)) and Ei = βi,1 S(−i − 1) for all 1 ≤ i ≤ p. In otherwords, property Np means the minimal free resolution of the homogeneous
coordinate ring S(X) of the projectively normal embedding of X is linear
until the p-th step.
In this talk, we are interested in geometric and algebraic properties ofa very ample subsystems of H 0 (L). There are two kinds of subsystems ofH 0 (L) with respect to projections. One gives an isomorphic outer projectionof X ⊂ P(H 0 (L)) with centers outside Sec(X) which is not linearly normal
and the other gives an inner projection of X which is closely related to thevery ample complete linear systems of blow ups of X. In their paper ([5]),S. Kwak and E. Park gave the geometric and syzygetic effects of property
Np of X ⊂ P(H 0 (L)) to the isomorphic projection of X in P(W ) by a very
ample subsystem W ⊂ H 0 (L). More precisely, one can generalize property
Np to property Np SW for a smooth variety X ⊂ P(W ) as follows: X satisfies
Lproperty Np SW if for R = `∈Z H 0 (X, L` ), it has the following minimal freeresolution of the simplest type as a graded SW -module;→ · · · → Ep → Ep−1 → · · · → E1 → SW ⊕ SW (−1)t → R → 0Lβi,1where SW = Sym(W ), t = codim(W, H 0 (L)) and Ei =SW (−i − 1)SW
for all 1 ≤ i ≤ p. In other words, property Npmeans the minimal freeresolution of R as a graded SW -module is linear until the p-th step. Recently,
Kwak and Park result is generalized to the following ([2]);12Theorem 1. Let X ⊂ Pr be a reduced nondegenerate projective subscheme
which satisfies property NpS . If X ⊂ Pr−t is an isomorphic linear projection
S .
where 0 ≤ t ≤ p, then X ⊂ Pr−t satisfies property Np−t
We are also interested in an inner projection of X which is given bysubsystem W of H 0 (L) passing through a given point q ∈ X. Let X be asmooth projective variety in P(H 0 (L)). For a closed point q ∈ X the innerprojection πq : X 99K P(W ) defined by πq (p) = qp∩P(W ), p 6= q, is a rational
map. We can understand this situation in the following diagram; for a blowup Blq (X) of X at q, one has the regular morphism πeq : Blq (X) → P(W )with a commutative diagram;
Blq (X) TTTTTTTTT π̃q
TTTTσTTTT*²/ X 0 = π (X \ {q}) ⊂ P(W )X ⊂ P (H 0 (L))πqq

It is said that X admits an inner projection at a point q ∈ X if themorphism πeq : Blq (X) → πq (X \ {q}) is an embedding, i.e. σ ∗ L − E is veryample. When one consider this situation in a projective embedding, we havea nice criterion for σ ∗ L−E to be very ample. Note that the inner projection
πeq : Blq (X) → P(W ) with center q ∈ X is a closed embedding if and only ifq∈
/ Trisec(X) where Trisec(X) is the union of all trisecant lines ` or ` ⊂ X([3]). In this case, πeq (Blq (X)) is equal to πq (X \ {q}), the Zariski closure ofπq (X \ {q}).
With these in mind, we get the following theorem([1]) for an inner projection;
Theorem 2. Let X ⊂ P(H 0 (L)) be a smooth irreducible projective variety.
Suppose L satisfies property Np for p ≥ 1. For any q ∈ X \ Trisec(X),πeq (Blq (X)) = πq (X \ {q}) in P(W ) is smooth and satisfies property Np−1 ,i.e. property Np−1 holds for (Blq (X), σ ∗ L − E).
Our method is cohomological, i.e. by using Koszul cohomology technique
due to M. Green, we can show TorSi W (R0 , C)i+j = 0, 0 ≤ i ≤ p and j ≥ 2 forLa finitely generated graded SW module R0 = `∈Z H 0 ((σ ∗ L − E)` ).
Finally, we talk about some applications of our results to various varieties
and their embeddings, for examples the inner projections of Veronese embeddings, Calabi-Yau manifolds, rational surfaces, and adjoint linear seriesof projective varieties.
References[1] Y.Choi, P. Kang, S. Kwak, Higher linear syzygies of inner projections and geometricapplications, preprint.

3[2] Y.Choi, S. Kwak, E. Park, Some effects of property Np , preprint.
[3] H. Flenner, L. O’Carroll, W. Vogel, Joins and intersections, Springer-Verlag, Berlin,(1999).
[4] M. Green, Koszul cohomology and the geometry of projective varieties, J.Differential
Geom. 19 (1984) 125–171.
[5] S. Kwak, E. Park, Some effects of property Np on the higher normality and defining
equations of nonlinearly normal varieties, J. Reine Angew. Math. 582 (2005) 87–105.