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\begin{document}

\title{Math 621: Topics in Algebraic Geometry}

\smallskip

\author{Juan Migliore \\
Math 621, Fall 2000}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcounter{puta}
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\newcounter{putd}


{\bf References and sources:}
\begin{enumerate}
\item Migliore's book
\item Pragmatic notes (Geramita and Migliore)
\item Migliore's Kyoto paper
\end{enumerate}

\bigskip

\centerline {{\bf Outline} }

\bigskip

\starta

\item Graded modules over a polynomial ring

\startb

\item Macaulay, CoCoA

\item $R = k[X{_0},...,X{_n}]$, $\proj{n}$

\startc

\item Noetherian

\stopc

\item shift  (and twist)

\item free modules

\item finitely generated modules

\startc

\item Examples

\startd

\item ideal

\item submodule of a free module (column space)

\stopd

\stopc

\item module structure


\startc

\item picture of multiplication

\item example: $k \oplus k$

\item represent structure by matrices of linear forms

\stopc

\item minimal generators

\item annihilator -- homogeneous ideal

\item finite length

\startc

\item diameter

\item Buchsbaum index ($\leq$ diameter)

\stopc

\item dual module

\stopb


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item Homogeneous ideals and schemes

\startb

\item dictionary -- see Cox-Little-O'Shea

\startc

\item prime ideals, radical ideals, etc.

\stopc

\item saturated ideals

\startc

\item example: compare the ideals $I_1 = (x,y)$, $I_2 = (x,y^2)$, $I_3 =
(x^2,xy,y^2)$, $I_4 = (x^2,y^2)$, $I_5 = (x^2,xy,xz,y^2,yz)$.

\item definition of saturated ideal, saturation

\item example from p.\ 2 of Fall 1992 -- ideal of 4 points in $\proj{3}$

\stopc

\item schemes

\item Hilbert function and polynomial

\startc

\item degree, arithmetic genus

\item example: $\proj{n}$ has dimension $n$, degree 1

\item example: a hypersurface is defined by a homogeneous polynomial of degree
$d$.  It has dimension $n-1$, degree $d$.

\stopc

\item linear systems, base locus

\item Examples

\startc

\item Fat points  (I)

\startd

\item Point of view: square the ideal and saturate.
Geometrically, we're looking for curves that are singular at all the points.

\item Example: three fat points lie (``unexpectedly'') on a cubic, namely the
union of the three lines that they span.

\item When is the square already saturated?
Example: one point, two points.  We'll see with liaison that if it's a complete
intersection then it's saturated.

\stopd

\item double line

\stopc

\item primary decomposition

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item minimal free resolutions -- follow Tony's notes.

\startb

\item   Goal: want the kernel to be free.  The kernel is
the module of $i$-th syzygies.

\item ``relations on the relations"

\item homological dimension

\item graded Betti numbers

\item Hilbert Szyzgy theorem

\item Castelnuovo-Mumford regularity (I)

\item criterion for minimality (give proof of both directions)

\item examples

\startc

\item first do $I = (x,y)$, then $I = (x,y,z)$

\item notice that it is a complex, but more precisely it is {\it exact}

\item do $I = (x^2,y^2,z^2)$ and then $I = (x^2,xy,y^2)$, first trying to
mimic the Koszul resolution, then giving the correct resolution

\item resolution for $k$

\stopc

\item Useful constructions

\startc

\item add trivial summand

\item direct sum resolution

\startd

\item Example: resolution for $k^2$

\item Example: resolution for $k \oplus k$ with trivial and nontrivial
structure

\stopd

\item Koszul resolution for a regular sequence

\stopc

\item Hom and Ext

\startc

\item motivation: a resolution is a complex, hence composition of two maps is
zero, hence composition of the transposes is also zero.  Want to make some
sense of this, by saying that if we go in the opposite direction we have at
least a complex, and $\Ext$ will measure how far we are from exactness.

\item Examples

\startd

\item $M = R(a) \Rightarrow \Hom_R(M,R) \cong R(-a)$

\item $M$ has finite length $\Rightarrow \Hom_R(M,R) = 0$.

\item Look at some of the complete intersections we've looked at: self-dual!

\stopd

\stopc

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item arithmetically Cohen-Macaulay schemes

\startb

\item Cohen-Macaulay rings

\startc

\item regular sequence

\item depth

\item dimension of a ring

\stopc

\item a zeroscheme is always ACM

\item Auslander-Buchsbaum

\startc

\item examples (also of non-ACM rings)

\stopc

\item projective dimension = codimension

\item Cohen-Macaulay type

\item arithmetically Gorenstein (I)

\startc

\item complete intersection, Koszul resolution

\item in codimension two CI $\Leftrightarrow$ arithmetically Gorenstein.  In
higher codimension this is not true.

\startd

\item Example: five points in $\proj{3}$.  Describe why, from maximal rank
point of view, we expect Gorenstein.

\stopd

\item We'll come back to it in much more detail
later.

\stopc

\item Minimal Resolution Conjecture (idea is the same as the maximal rank idea
that just gave us Gorenstein for 5 points in $\proj{3}$).

\startc

\item generic Hilbert function

\stopc

\item canonical module

\startc

\item resolution of the canonical module for ACM schemes

\item duality for arithmetically Gorenstein schemes

\item Prop 4.1.1, Cor 4.1.3 of my book

\stopc

\item Hilbert function of points

\item truncation

\startc

\item basic lemma

\item DGM -- will talk in seminar

\stopc

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item Sheaves and sheaf cohomology

\startb

\item basics about sheafification-- see PRAGMATIC notes

\item Sheafification of a short exact sequence of graded modules

\item Sheafification of a resolution

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item Schemes and Deficiency Modules

\startb

\item deficiency modules (definition)

\item Relation between Ext groups and cohomology of the ideal sheaf. (See
Schwartau Lemma 31, then generalize it.  See also BM4.)

\startc

\item sketch proof for surface in $\proj{5}$.

\item Corollary: ACM $\Leftrightarrow$ deficiency modules are zero.

\item Give second proof without Ext's.

\stopc

\item Examples

\startc

\item two skew lines

\item disjoint union of a line and a conic

\item disjoint union of a line and a plane curve

\item maximal rank curve, e.g. general set of skew lines
(Hartshorne-\break Hirschowitz), general rational curve (Hirschowitz).  Mention
Ballico-Ellia.

\stopc

\item Castelnuovo-Mumford regularity (II)

\startc

\item examples

\stopc

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item Hyperplane and Hypersurface sections

\startb

\item introduction:  depth $R/IV = 1$ iff every $\bar F_2$ is a zero divisor in
$R/(I,F_1)$ (for any choice of $F_1$ which is not a zero divisor in $R/I_V$),
and this is true iff $(I_V,F_1)$ is not saturated.  So we want to study ideals
of this form.

\item algebraic, geometric hyperplane or hypersurface section

\item exact sequences that arise -- see Pragmatic notes and book

\startc

\item Corollary: $K_F$ measures the failure of $I_V + (F)$ to be saturated.

\item Corollary: $I_V + (F)$ is saturated iff $(M^1)(V) = 0$.  (Use Serre's
theorem, [Hartshorne] p.\ 228.)

\stopc

\item Mention Dubreil result.

\item What's preserved under {\em geometric} hyperplane sections?

\startc

\item dimension is one less

\item the ACM property is preserved.

\item In fact, the converse holds if stated properly.  Note need dimension
$\geq 2$.

\startd

\item Example of projection of Veronese.

\item Say a few words about the curve case, mentioning Strano, Re, H-U, me.

\stopd

\stopc

\item Hilbert function, first difference

\item Artinian reduction of an ACM scheme

\startc

\item Hilbert function is the first difference: $h$-vectors

\stopc

\item graded Betti numbers preserved in aCM case

\item CoCoA, macaulay

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\item Deficiency modules revisited

\startb

\item Note structure is important to specify the module,  not just the
dimension of the components.  This is the ``hard'' part.  Depends a lot on the
geometry of the curve (or scheme).

\item Example: disjoint union in $\proj{3}$ of a line and a conic.

\item Philosophy (state in the case of curves, maybe higher too): those linear
forms having ``unusual'' rank on the HR-module correspond to planes meeting the
curve in ``unusual'' ways, either by containing a component of the curve or by
having unusual postulation for the hyperplane section.

\item What can happen in negative degrees to $h^1({\mathcal I}_C (t))$ and $h^2
({\mathcal I}_C (t))$?

\startc

\item If $C$ is reduced then

\startd

\item $M(C)_i = 0$ for $i < 0$.

\item $M(C)_0 =$ (number of connected components of $C$) - 1.

\stopd

\item Rao module dimensions are non-decreasing.

\item In fact dimensions are strictly increasing.

\stopc

\item General question: which shifts occur?

\startc

\item Theorem: for any $M$, if you shift far enough to the right then curve
exists.  (Just state for $\proj{3}$.)  Sketch proof.

\item Stress that it's not true without shifting.

\item Now turn to a related question, with direct sums.

\stopc

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\item Liaison Addition

\startb

\item Remark about how all modules occur, so Phil's is a natural question.

\item Note first question is false: Take both curves to be disjoint union of
two lines.

\item Schwartau's result (state and prove for curves in $\proj{3}$).

\item give geometric interpretation.

\item Generalized result (just state, and mention that the proof is very
similar with a few small complications)

\item Application: Basic double linkage.  (We'll see where the name comes from
later.)

\item Basic double G-linkage (1st visit).

\item Application: construct examples of schemes with embedded components

\item Construct Buchsbaum curves

\startc

\item $k^2$, $k \oplus k$ (Berlin picture)

\item mention both of these satisfy $\nu = 3N+1$, $\alpha = 2N$.

\stopc

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item  Buchsbaum subschemes of $\proj{n}$, especially Buchsbaum curves
in $\proj{3}$

\startb

\item Definition of Buchsbaum curve.

\item Definition of Buchsbaum subscheme.

\item Rao's result: relation between resolution of curve and module

\item See how much is forced on a curve by knowing that its module has trivial
structure: BSV result, Amasaki results.

\item GM1: show what shifts can occur for Buchsbaum curves in $\proj{3}$.

\item connected except for 2 skew lines

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item Gorenstein Schemes and Ideals

\startb

\item Recall definition

\item Constructions and Theorems

\startc

\item sums of G-linked ideals

\item Linear system construction

\item Sections of Buchsbaum-Rim sheaves of odd rank.

\stopc

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item  Liaison: Definitions, Examples, Questions

\startb

\item history

\item {\it liaison} (start off in arbitrary codimension)--- intuitive
definition, geometric liaison, algebraic liaison, precise definition.

\item geometric link $\Rightarrow$ algebraic link, but not converse.

\item We'll see soon that algebraic link and no common component $\Rightarrow$
geometric link.

\item ${I_X}:{I_V}$ is saturated


\item even liaison, {$\mathcal L$} = even liaison class

\item Questions:

\startc

\item Find connections between the linked schemes (degree, genus, and
especially more subtle connections)

\item Do geometric and algebraic links generate the same equivalence relation?

\item Is this a trivial equivalence relation?  Mention aCM example, especially
in codimension two.  Also the difference between codim 2 and the general case.
(In particular, in $\proj{2}$ codimension 2 it {\it is} trivial, and in
codimension
1 anywhere, but not otherwise.)

\item Parameterize the (even) liaison classes--- give necessary and
sufficient conditions for two subschemes to be in the same (even) liaison class

\item Describe any one even liaison class--- {\it structure}--- in
particular, are they all the same?

\item Applications?

\stopc

\item Examples

\startc

\item CoCoA

\item twisted cubic $\sim$ line

\item concocted example where residual is non-reduced

\item mention that sum of degrees of linked schemes = degree of the
complete intersection

\item rational sextic with a 5-secant line--- linking with two cubics
{\it forces} the residual to be non-reduced.  (Need to show that there's
exactly one 4-secant in addition to the 5-secant.)

\item self-linked

\stopc

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item  First results (arbitrary codimension)

\startb

\item Any two complete intersections (of the same dimension) are linked--- for
example see Phil's Theorem 14.

\startc

\item definitions of licci, glicci

\item Casanellas- Mir\'o-Roig result for gorensteins being glicci

\stopc

\item linearly equivalent $ \Rightarrow$ evenly linked in two steps

\item Choose $V_1$ and $X \Rightarrow$ get ${V_2} {\buildrel X\over \sim}
{V_1}$

\item For curves, non locally CM $\Rightarrow$ singleton class.  So restrict to
locally CM.

\item If $V_1 {\buildrel X\over \sim} {V_2}$ then $V_1 ,V_2$ are
equidimensional of the same dimension, and without embedded components (see
Phil p. 28).

\item Linked scheme is again locally CM of the same dimension (Phil Theorem 15)
-- follow my book prop.\ 5.2.2, prop.\ 5.2.3.

\item dualizing sheaf exact sequence

\item Mapping Cone

\startc

\item Corollary: for aCM codimension two, linking using two (resp. one,
zero) minimal generators drops the number of minimal generators for the
residual
by one (resp. leaves it the same, increases it by one).  (Mention
Hilbert-Burch.)

\stopc

\item degree, genus, Hilbert function of residual

\item Examples

\startc

\item twisted cubic $\sim$ line again

\item two things of same degree linked $\Rightarrow$ same genus

\stopc

\item Are hyperplane sections linked?  Yes.

\item Necessary conditions--- Hartshorne-Schenzel

\startc

\item Note that for dimension $\geq$ 2, the {\it configuration} of modules is
preserved up to shift.

\item Useful for showing things are {\it not} linked (as in next ``chapter'')

\item Corollary: necessary condition for self-linked

\item Corollary: the property of being Buchsbaum or aCM is preserved under
liaison.

\stopc

\item Theorem: Ap\'ery-Gaeta-Peskine-Szpir\'o. (codimension two) The
part that's a corollary is that {\it licci} $\Rightarrow$ arithmetically CM.
For the other direction, use the fact that linking using minimal generators
drops the number of generators.  For this it'll be necessary to talk about
Hilbert-Burch stuff

\item up to shift, all modules occur (Schenzel, Rao)

\item see why even liaison is better

\item Minimal shift

\startc

\item Lemma: any rightward shift occurs

\item Leftward shifts in general

\stopc

\item Special case: Curves

\startc

\item Recall what can happen in negative degrees to $h^1({\mathcal I}_C (t))$
(non-decreasing)

\item Minimal shift exists (and all subsequent shifts occur)

\stopc

\item In general: notion of shift (partition class), minimal shift, notation
${\mathcal L}^h$

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item  Geometric Invariants of Liaison (especially curves in $\proj{3}$)

\startb

\item How do you describe the module structure? $\phi_d : S_1 \rightarrow
\hbox{Hom}(M_d , M_{d+1} )$, then describe it in terms of matrices.

\startc

\item Example-- diameter two, dimensions 1,1.  Discuss the possible
isomorphisms from this point of view.

\item Mention Ballico-Bolondi, $\nu_{{\bf P}^n} (m_1 ,...,m_t )$

\item Example-- diameter 3, dimensions 1,1,1.  Possible isomorphisms.

\stopc

\item Definitions of the degeneracy loci

\startc

\item Porteous' Formula

\stopc

\item Some results about when you know that a linear form corresponds to a
point of the degeneracy locus.

\startc

\item standard result (postulation of hyperplane section)

\item some cases when the hyperplane contains a component of the curve.

\stopc

\item Examples

\startc

\item disjoint union of a line and a conic

\item disjoint union of a line and
a  plane curve.  Note about degeneracy loci being all the same.  (To see
that two
such curves with same degeneracy locus are linked, have to link directly.  But
note that modules are isomorphic anyway.)

\item Buchsbaum curves: no degeneracy loci

\item example on page 44 of thesis

\item skew lines ($\proj{3}$ and $\proj{4}$), including case on a quadric.
Note
that the results in $\proj{4}$ are very similar to those in $\proj{3}$.

\item curves of low degree, especially rational sextic.  (Degeneracy locus
isn't enough for isomorphism.)

\stopc

\item Lazarsfeld-Rao ``e+4 $\Rightarrow$ recover $C$'' result (generalize skew
lines)

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item  Sufficient conditions in Codimension Two

\startb

\item Rao's main result (Math. Ann. paper)

\item Corollary: curves in $\proj{3}$

\item Questions

\startc

\item Is there a module analog for Rao's theorem (codimension 2)?

\item What is the right theorem in higher codimension?

\item Can we even find an example, apart from aCM, of two schemes with the same
modules but lying in different even liaison classes?  (Conjecture: two skew
lines in $\proj{4}$--- recall results about liaison of lines in $\proj{4}$.)

\item Can we find an example of two surfaces in $\proj{4}$ which are in
different even liaison classes, but for a general hyperplane the corresponding
curves are evenly linked?  Yes: use the surface from Decker-Ein-Schreyer which
is quasi-Buchsbaum but the hyperplane section has module of type (1,1) but not
Buchsbaum, and for the second surface maybe take a cone over the disjoint union
of a line and a conic.

\stopc

\item Examples

\startc

\item double lines

\item ${\mathcal L}^h_{n{_1},...,n{_t}}$

\startd

\item Recall results about shifts

\stopd

\item disjoint union of a line and a plane curve

\stopc

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item  Structure of an even liaison class in codimension two

\startb

\item Basic double links (special case of liaison addition)

\item How to get the deformations

\startc

\item relation to Hilbert function and shift

\item Ballico-Bolondi results

\stopc

\item Lazarsfeld-Rao, BM3 (no proof since these are now special case of BBM)

\item Ballico-Bolondi-Migliore

\item Results from BM4, BM5:

\startc

\item things about maximal rank

\item do basic double links in increasing degree

\item partition of even liaison class (mention Ballico-Bolondi for
structure of family)

\item single liaison addition

\stopc

\item Martin-Deschamps-Perrin (curves in $\proj{3}$)

\item Bolondi (surfaces in $\proj{4}$)

\stopb

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item  Applications

\startb

\item Maggioni-Ragusa

\item Mir\'o-Roig

\item stick figures (BM5)

\stopb

\stopa
\end{document}
