Gregory Sankaran: Research
Department of Mathematical Sciences, University of Bath
This page lists and describes my research activities. For
undergraduatelevel and other mathematics, see
my teaching page; for general
interests, my personal page; or for
directions, my home page.
Mathematicians' personal web pages are expected to have a picture of
them in Oberwolfach. This one was taken by Slava Nikulin.
 Papers
 Research students
 Other research activities
Most of my papers since 1992 can be found on
the arXiv: all of them are at least
crossposted to math.AG. The list below is more
comprehensive, though. It includes some papers too old for arXiv, some
that are ineligible because they are purely surveys or in the nature
of technical reports, and some that I chose not to publicise that way.

Rationality and arithmetic of the moduli of abelian varieties
[ps]
[pdf]
Daniel Loughran and Gregory Sankaran.
Establishes the unirationality (for g=4,5) and stable
rationality (for g=3) over the rational numbers of the moduli
space of principally polarised abelian gfolds. This has
entertaining consequences for arithmetic.

Slopes of Siegel cusp forms and geometry of
compactified Kuga varieties
[ps]
[pdf]
F. Poon, R. Salvati Manni and G.K. Sankaran.
Determines the Kodaira dimension of all the nfold families of
Kummer varieties over the moduli space of principally polarised
abelian gfolds.

Lazardstyle CAD and Equational Constraints
[ps]
[pdf]
James Davenport, Akshar Nair, Gregory Sankaran and Ali Uncu.
ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation, July 2023, pp 218226.
doi: 10.1145/3597066.3597090

The CAD Conundrum: LexLeast vs Order
[ps]
[pdf]
S. McCallum, A. Nair, J. Davenport and G. Sankaran. 2020
22nd International Symposium on Symbolic and Numeric Algorithms for
Scientific Computing (SYNASC), Timisoara, Romania, 2020, pp. 3235. doi: 10.1109/SYNASC51798.2020.00017

A supersingular coincidence
[ps]
[pdf]
G.K. Sankaran. Ramanujan J. 59 (2022) 609613.
doi: 10.1007/s11139021005264
Explains why the
fifteen supersingular primes
are the ones for which it is known that the moduli space of
(1,p)polarised abelian surfaces is of general type.

Blowups with log canonical singularities
[ps]
[pdf]
G.K. Sankaran and F. Santos. Geometry & Topology 25 (2021), 21452166.
Proves a conjecture of Caucher Birkar, using toric geometry
and lattice polytopes.

Curtains in CAD: why are they a problem and how do we fix them?
[ps]
[pdf]
A.S. Nair, J.H. Davenport and G.K. Sankaran.
Fuller version of a decomposition algorithm for a hypersurface.

Locally symmetric varieties and holomorphic symplectic manifolds
[ps]
[pdf]
G.K. Sankaran.
Notes from a survey talk in Kinosaki, October 2019.

Saturated and primitive smooth compactifications of ball quotients
[ps]
[pdf]
P.G. Beshkov, A.K. Kasparian and G.K. Sankaran. Ann. Sofia Univ. Fac. Math. and Inf. 106 (2019), 5377.
A progress report: contains some partial results, published at this
stage mainly to comply with Bulgarian PhD regulations.

Lazardâ€™s CAD exploiting equality constraints
[ps]
[pdf]
A. Nair, J.H. Davenport, G.K. Sankaran and S. McCallum. ACM
Communications in Computer Algebra 53 No. 3 (2019),
138141.
Modifies a standard algorithm for decomposing real affine space so as
to decompose a hypersurface instead.

On benefits of equality constraints in lexleast invariant CAD
(extended abstract)
[ps]
[pdf]
A.S. Nair, J.H. Davenport and G.K. Sankaran. In: Proceedings SC2 2019.
Provides some more detail of the decomposition algorithm.

Toroidal compactification: the generalised ball case
[ps]
[pdf]
A.K. Kasparian and G.K. Sankaran. In: Moduli Spaces and Locally Symmetric Spaces (Lizhen Ji and ShingTung Yau, Eds.), Surveys in Modern Mathematics 16, Ch. 3, pp. 107133, Higher Education Press, Beijing 2021.
Largely a survey: a worked example of toroidal compactification, with
little claim to originality but perhaps useful.

Fast Matrix Operations in Computer Algebra
[ps]
[pdf]
Z. Tonks, J.H. Davenport and G.K. Sankaran. In: T. Jebelean,
V. Negru, D. Petcu, D. Zaharie, T. Ida and S. Watt,
Eds., Proceedings 19th International Symposium on Symbolic and
Numeric Algorithms for Scientific Computing (SYNASC 2017), Timisoara,
Romania, 2124 September 2017, pp. 6770.
A little elementary algebra applied to a small practical computing
problem. Notable mainly for citing Dodgson (that is, Lewis Carroll) in
a genuine mathematical context.

Regular Cylindrical Algebraic Decomposition [ps] [pdf]
J.H. Davenport, A.F. Locatelli and G.K. Sankaran. J. London Math. Soc. 101 No. 1 (2020), 4359.
Topology and real semialgebraic geometry: a harmless problem coming
from computer science meets the hcobordism theorem.

Weighted CastelnuovoMumford regularity and weighted global
generation
[ps]
[pdf]
F. Malaspina and G.K. Sankaran. J. Algebra
Appl. 18 No. 2 (2019), 1950028.
A failure report on an attempt to define a notion of regularity
adapted to weighted projective spaces. It does work, but not very
well.

Genus 4 curves on abelian surfaces [ps] [pdf]
P. Borówka and
G.K. Sankaran. Proc. Amer. Math. Soc. 145 No. 12
(2017), 50235034.
Surprisingly little is known about this sort of question. The
hyperelliptic curves are particularly interesting.

Fundamental groups of toroidal compactifications [ps] [pdf]
A.K. Kasparian and G.K. Sankaran. Asian
J. Math. 22 No. 5 (2018), 941954.
We tell you what the fundamental group of a toroidal compactification
is, in terms of the arithmetic group involved.

Moduli of K3 surfaces and irreducible symplectic manifolds [ps] [pdf]
V. Gritsenko, K. Hulek and G.K. Sankaran. In: G. Farkas &
I. Morrison (Eds.), Handbook of Moduli, Vol I,
Adv. Lect. Math. (ALM) 24 Int. Press, Somerville, MA (2013),
459526.
A survey article, fairly thorough and including some new material, on
moduli of polarised hyperkähler manifolds.

On some lattice computations related to moduli problems [ps] [pdf]
A. Peterson and G.K. Sankaran, with an appendix by V. Gritsenko.
Rend. Sem. Mat. Univ. Pol. Torino 68 No. 3 (2010), 289304.
This fills a small gap in the paper on K3 surfaces: it settles the
case d=52 which should have been mentioned there, but wasn't.
There is a supplement
containing the computer programs used in this paper.

Moduli spaces of polarised symplectic O'Grady varieties and
Borcherds products
[ps]
[pdf]
V. Gritsenko, K. Hulek and G.K. Sankaran.
J. Differential Geometry 88 No. 1 (2011), 6185.
Handles one of the known types of hyperkähler manifold (the OG10 case).

Smooth rationally connected threefolds contain all smooth curves [ps] [pdf]
G.K. Sankaran. In: W. Ebeling, K. Hulek & K. Smoczyk
(Eds.), Complex and Differential Geometry, Springer Proceedings
in Mathematics 8 (2011), 393402.
This "cool result" (Richard Thomas) answers a question of Eisenbud and
Harris but does not appear to be of any actual use.

Boundedness for surfaces in weighted projective 4spaces [ps] [pdf]
L.V. Rammea and G.K. Sankaran.
Proc. Amer. Math. Soc. 139 (2011), 33933403.
Any quasismooth surface in any weighted projective space of
sufficiently high degree is of general type.

Abelianisation of orthogonal groups and the fundamental group of
modular varieties [ps] [pdf]
V. Gritsenko, K. Hulek and G.K. Sankaran.
J. Algebra 322 (2009), 463478.
Technicalities about the structure of arithmetic subgroups of
O(2,n).

Moduli spaces of irreducible symplectic manifolds [ps] [pdf]
V. Gritsenko, K. Hulek and G.K. Sankaran.
Compositio Math. 146 (2010) 404434.
Foundational paper on the moduli of polarised hyperkähler
manifolds.

The moduli space of étale double covers of genus 5 curves is
unirational [ps] [pdf]
E. Izadi, M. Lo Giudice and G.K. Sankaran.
Pacific J. Math. 239 (2009), 3952.
Surprisingly this was not known.

Numerical obstructions to abelian surfaces in toric Fano 4folds [ps] [pdf]
G.K. Sankaran.
Kodai Math. J. 31 (2008), 120.
This corrects the
error in Abelian surfaces in toric 4folds, below, and locates
the cohomology classes in smooth toric Fano 4folds that could be
represented by an abelian surface.

HirzebruchMumford proportionality and locally symmetric varieties
of orthogonal type [ps] [pdf]
V. Gritsenko, K. Hulek and G.K. Sankaran.
Documenta Math. 13 (2008), 119.
Systematically exploits the asymptotic approach to estimating
plurigenera of orthogonal locally symmetric varieties.

Cusp forms: a clarification [ps] [pdf]
G.K. Sankaran.
Unpublished note, 2008.
This fills a small gap, pointed out by
R. Salvati Manni, in The Kodaira dimension of the moduli of K3
surfaces below, and is put here for reference. The matter is
more fully addressed in later papers.

The Kodaira dimension of the moduli of K3 surfaces [ps] [pdf]
V. Gritsenko, K. Hulek and G.K. Sankaran.
Invent. Math. 167 (2007), 519567.
The
original publication is available at
www.springerlink.com.
Almost all moduli spaces of polarised K3 surfaces are of general type.

The HirzebruchMumford volume for the orthogonal group and
applications [ps] [pdf]
V. Gritsenko, K. Hulek and G.K. Sankaran.
Documenta Math. 12 (2007), 215241.
Numbertheoretic technicalities associated with O(2,n)

The nef cone of toroidal compactifications of
A_{4} [ps] [pdf]
K. Hulek and G.K. Sankaran.
Proc. London Math. Soc. 88 (2004), 659704.
A technical look at the geometry of the 2nd Voronoi compactification
of the moduli of principally polarised abelian 4folds.

Abelian surfaces with odd bilevel structure
[ps] [pdf]
G.K. Sankaran.
In: M. Reid, A. Skorobogatov (Eds.), Number theory and algebraic
geometry, London Math. Soc. Lecture Note Series,
303 (2003), 279300.
A study of a special moduli space of abelian surfaces introduced by
Mukai.

The moduli space of bilevel6 abelian surfaces [ps] [pdf]
G.K. Sankaran and J. Spandaw.
Nagoya Math. J. 168 (2002), 113125.
A special case with some extra geometry.

Das Titsgebäude von Siegelschen Modulgruppen vom Geschlecht
2 [ps] [pdf]
M. Friedland and G.K. Sankaran.
Abh. Math. Sem. Univ. Hamburg 71 (2001), 4968.
A tedious but at the time useful computation, written German because
half of it was taken from Friedland's Diplomarbeit.
There is a supplement with an
English summary [ps]
[pdf]
and some pictures. The pictures are quite pretty.

Algebraic construction of normalized coprime factors for delay
systems [ps] [pdf]
J.R. Partington and G.K. Sankaran.
Math. Control Signals Systems 15 (2002), 112.
I know nothing about this subject.

The geometry of Siegel modular varieties [ps] [pdf]
K. Hulek and G.K. Sankaran. In: S. Mori, Y. Miyaoka (Eds.),
Higher Dimensional Birational Geometry,
Advanced Studies in Pure Mathematics 35 (2002), 89156.
A survey article, with some new material: apparently still not out of date.

Abelian surfaces in toric 4folds [ps] [pdf]
G.K. Sankaran.
Math. Ann. 313 (1999), 409427.
An attempt to understand this problem, but the example in the last
section is completely wrong because of a simple miscalculation. The
idea is all right.

Degenerations of (1,3) abelian surfaces and Kummer surfaces [ps] [pdf]
K. Hulek, I. Nieto and G.K. Sankaran.
In: P. Pragacz, M. Szurek, J. Wisniewski (Eds.),
Algebraic Geometry: Hirzebruch 70,
AMS Contemporary Mathematics 241 (1999), 177192.
Details of how polarised abelian surfaces degenerate.

Heisenberginvariant Kummer surfaces [ps] [pdf]
K. Hulek, I. Nieto and G.K. Sankaran.
Proc. Edin. Math. Soc. 43 (2000), 425439.
Some special geometry, eventually reaching back into early 20th
century English projective geometry.

Recent results on moduli of abelian surfaces
[ps]
[pdf]
G.K. Sankaran.
Notes from a survey talk in Kinosaki, November 1993.
I have two current research students:
 Flora Poon started in 2020. She works in the general area of locally symmetric varieties and moduli of curves, K3 surfaces and abelian varieties.
 Corin Lee started in 2021. He is supervised by James Davenport
and me jointly. He works on theory and algorithms for
algebraic decomposition of semialgebraic sets.
Eleven people have so far completed a PhD under my supervision.
 Alfio Marini
worked on polarised abelian surfaces and graduated in 2002.
Alfio
Marini's
thesis
as Postscript
or
PDF.
 Al
Kasprzyk worked on toric geometry and polytopes and graduated in
2006.
Al Kasprzyk's thesis
as Postscript
or PDF.

Marco Lo
Giudice worked on moduli of Pryms and graduated in 2006 with a
Ph.D. at the Università di Milano.
Marco Lo Giudice's
thesis
as Postscript
or
PDF.

Nathan
Broomhead worked on dimer models and graduated in 2009. He was
supervised by
Alastair King and me
jointly.
Nathan Broomhead's thesis as Postscript or
PDF.

Lisema
Rammea worked on surfaces in weighted projective 4spaces and
graduated in 2009.
Lisema Rammea's thesis
as Postscript
or
PDF.

Paweł
Borówka worked on special curves in abelian surfaces and
graduated in 2012.
Paweł Borówka's thesis
as Postscript
or
PDF.

Acyr Locatelli
worked on the topology of cylindrical algebraic decomposition and
graduated in 2015. He was supervised by
James Davenport
and me jointly.
Acyr Locatelli's thesis as Postscript or
PDF.

Matthew Dawes
worked on moduli of generalised Kummer manifolds and graduated in
2015.
Matthew Dawes' thesis
as Postscript
or
PDF.

Claudio
Onorati worked on families of OG10 manifolds and graduated in
2018.
Claudio Onorati's thesis as
Postscript
or
PDF.
 Akshar Nair worked on efficient algorithms for cylindrical
algebraic decomposition and real algebraic geometry, and graduated
in 2021. He was supervised
by James Davenport
and me jointly.
Akshar Nair's thesis as Postscript or
PDF.

Calla Tschanz worked on Hilbert schemes and their degenerations, and hyperkähler manifolds, and graduated in 2023.
Calla Tschanz's thesis as
Postscript
or
PDF.
Some other people should be mentioned here. Atika Ahmed started a PhD
with me in 2011, but for health reasons she was not able to complete
it. Timothy
Logvinenko studied with Alastair King and graduated in 2004, but
for a while I was officially also supervising him, for technical
reasons. I don't claim any credit, but I want to acknowledge the link.
Here are some other mathematical things that I do or have done. They
may give you some idea of what I can be persuaded to do.
 Organising COW, 19922018.
 Part (small part) coordinating EAGER, circa 2002. It was a
European training network, and it was funded for pure abstract
research. Those were the days.
 LMS Editorial Advisory Board, 20032013.
 EPSRC College, since 2014.
 EPSRC Mathematical Sciences Strategic Advisory Team, since
February 2020.
 MARM (Mentoring African Research in Mathematics) at
Makerere
University, Kampala.
 Also to do with Africa: the Abram Gannibal Project
 Programme Committee of ICMS
Edinburgh.
 An Academic Adviser to the Commonwealth Scholarship Commission.
 Assorted conference organisation.
 Scientific committees of a few conferences. One of those was
BrAG, until they threw me off so that they could invite me to give a talk. Being on the Scientific Committee usually involves no work.
 PhD external examining for lots of places. Depending on the
system this may involve little work or a lot.
 Research visits to lots of places.