# Roy Smith

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Professor Emeritus, Years at UGA: 1977-2010

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My research interests are in Algebraic Geometry.

Here are my current preprints, all joint with Robert Varley:
1) A Torelli theorem for special divisor varieties associated to doubly covered curves, sv1nr.pdf

2) A Riemann singularities theorem for Prym theta divisors, with applications, sv2rst.pdf

3) The curve of Prym canonical Gauss divisors on a Prym theta divisor, sv3pg.pdf

4) The Prym Torelli problem: an update and a reformulation as a question in birational geometry,  sv4ut.pdf

5) A necessary and sufficient condition for Riemann's singularity theorem to hold on a Prym theta divisor, sv5rst2.pdf

6) The Pfaffian structure defining a Prym theta divisor, sv6pfaff.pdf

7) Deformations of isolated even double points of corank one, sv7cork1defs.pdf

8) A splitting criterion for an isolated singularity at x = 0 in a family of even hypersurfaces, sv8poscrkdefs.pdf

9) On parametrizing exceptional tangent cones to Prym theta divisors, onparam.pdf

Here are some class notes. Take whatever you like.

1. (rev.lin.alg.pdf):  Linear algebra notes, including spectral theorem for symmetric operators, jordan form, rational canonical form, minimal and characteristic polynomials, and Cayley Hamilton, all in 15 pages!

2. (RRT.pdf):  A discussion of the easy aspects of the Riemann Roch theorem for curves, surfaces, and n dimensional smooth manifolds. We give Riemann's classical proof for curves, assuming his results on the existence of meromorphic differentials of first and second kinds. Then we reprove from scratch the Hirzebruch version for curves, using (and recalling) sheaf cohomology, but only sketching Serre's proof of the duality theorem. Then we prove similarly Hirzebruch's version for smooth surfaces embedded in projective three space. Finally we sketch the formalism of chern roots and their use in defining Todd classes and in stating the general HRR. This is aimed at a grad student who has had complex analysis of one variable, and a little topology.

3. Algebra course notes

a. 843-1.pdf,  Groups, and groups acting on sets.

b. 843-2.pdf,  Why some polynomials cannot be solved by radicals.

c. 844-1.pdf, Rings, fields, and the Galois correspondence between subgroups of automorphisms, and subfields of extensions.

d. 844-2.pdf, G(K/L) solvable implies (K/L) radical (in char.zero); examples where G = S(3), S(4), any finite abelian group.

e. 845-1.pdf, Decomposing f.g. modules over a pid, by diagonalizing a presentation matrix; e.g. min'l polynomial, "rat'l canonical" form.

f. 845-2.pdf, Existence of "Jordan" form of a linear operator if min'l polynomial has all factors linear; existence of "diagonal" form if min'l polynomial has all factors linear and distinct, or if the matrix is "symmetric" /R or "normal"/C.

g. 845-3.pdf, Hom functors, duality, tensor products, exterior products.

4. Elementary Algebra course notes.

a. 4000.01-05.pdf    Well ordering, induction, binomial thm., Euclidean algorithm, gcd's, infinitude of primes.

b. 4000.06-09.pdf   Modular arithmetic, solving congruences, Fermat little theorem, rings, domains, fields

c. 4000.10-13.pdf  rational numbers, irrational numbers, rational roots theorem, real numbers, infinite decimals, geometric series and repeating decimals, adjoining square roots to Q

d. 4000.14-20.pdf   complex numbers and trigonometry, complex rationals, complex ("Gaussian") integers, Gaussian primes and Fermat's theorem on sums of 2 squares.

e. 4000.21-24.pdf   polynomials, division algorithm, modular polynomial rings, connection with fields obtained by adjoining roots.

f. 4000.25-30.pdf  vector dimension of field extensions, multiplicativity of dimension, dimension of root fields, dimension of constructible extensions, impossibility proofs.

5. Algebraic Geometry Notes.

a. introAG.pdf Naive introduction to algebraic geometry.

b. Riemann.pdf A sketch of Riemann's approach to classifying convergent power series.

6. Review for PhD prelim preparation in algebra (100 pages)

These condensed notes include basic theorems about pid: uniqueness of factorization and decomposition of finitely generated modules, application to Jordan and rational canonical forms of matrices; also Gausstheory of content and unique factorization in Z[X], Dumas Eisenstein irreducibility, Noetherian rings, Sylow theorems, Jordan - Holder, the fundamental theorem of Galois theory, Zorn lemma, existence of algebraic closures of fields, normality, separability, cyclotomic polynomials, insolvability of general polynomials of degree > 4, duality and spectral theorems. These notes are most useful for someone reviewing the material a second time. Consult the 843-4-5 course notes for more details, examples and omitted topics (tensors).

a.8000a.pdf course outline,

b.8000b.pdf ab grps, rings, modules,

c.8000c.pdf linear algebra,

d.8000d.pdf grps, fields galois,

e.8000e.pdf hw, tests

4050sum08.pdf

Review of basic definitions, dimension theory, statements of basic facts about row reduction, and detailed proofs of existence and uniqueness of jordan forms for split minimal polynomials, as well as generalized jordan forms (rational canonical forms) for all minimal polynomials.

Every decomposition theorem is proved three times, in increasing degree of complexity: i) reduced minimal polynomial, i.e. maximal number of cyclic factors, or diagonal case, ii) minimal polynomial equals characteristic polynomial, i.e. minimal number of factors, or cyclic case, iii) general case.

There is also a sketch of existence of cyclic product decomposition for finite abelian groups, a complete treatment of determinants and the cayley- hamilton theorem. a complete treatment of "spectral" i.e. structure theorems for normal operators in finite dimensional real and complex spaces, some discussion of duality, solving homogeneous ode's with constant coefficients to motivate jordan form, of inverting linear constant coefficient operators "locally" on spaces of polynomials, and a definition of the derivative for any locally integrable function as an adjoint operator on distributions.

8.On teaching.

9. Math 5200: metric Euclidean geometry

a. 5200.1 axioms, congruence fa 07.pdf

b. 5200.2 parallel lines.pdf

c. 5200.3 fa07 similarity 1.pdf

d. 5200.4 fa07 euclid's content 1.pdf

e. 5200.5 fa07 law of cosines.pdf

f. 5200.6 fa07 concurrence.pdf

10. Euclidean geometry

11. Math 2200 differential calculus

Woods Hole conference 1964

Some people have expressed a wish to have a copy of the proceedings of the 1964 conference in algebraic geometry at Woods Hole, Massachusetts.
Here is an apparently complete set, courtesy of my friend Doug Clark, who attended the meeting.

woods hole 1.  Theory of singularities: Abhyankar, Hironaka, Zariski;
Classification of surfaces and moduli: Kodaira, Matsusaka, Mumford, Nagata, Rosenlicht, Igusa.

woods hole 2. Grothendieck cohomology: Artin, Verdier, Tate.
Zeta functions and arithmetic of abelian varieties: Cassels, Dwork, Shimura, Serre.

woods hole 3. Seminar on singularities: Abhyankar, Hironaka, Zariski.
Moduli questions: Ehrenpreis, Kodaira, Mayer, Mumford, Rauch.
Seminar on etale cohomology of number fields: Artin, Verdier.
Families of abelian varieties and number theory: Kuga, Shimura.
Seminar on commutative algebra: Auslander, Greenleaf, Lichtenbaum, Rim, Samuel, Schlessinger.
Etale cohomology: Hartshorne.
Fixed point theorem seminar: Atiyah, Bott.

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