Alice Rizzardo

SISSA
Area Matematica

Office: 719
Phone: (+39) 040 3787856
Email: arizzardo@sissa.it
My Curriculum Vitae.


I am a Postdoctoral Research Fellow at SISSA. I graduated from Columbia University in May 2012. My advisor was Johan de Jong. I have held postdoctoral positions at the School of Mathematics of the University of Edinburgh and at MSRI.

Starting in October 2016 I will be an EPSRC Postdoctoral Fellow at the University of Edinburgh. Here is the abstract of my proposal.

I am an algebraic geometer. My current research involves understanding algebraic varieties through the lens of their derived category. This leads naturally to investigating the behavior of derived categories and dg-categories using techniques ranging from homological algebra to representation theory. Much of my work has been devoted to studying functors between derived categories of coherent sheaves on projective varieties, and how they can be expressed in a geometric way.

Upcoming Talks
GVA2016, Geometry and Classification of Algebraic Varieties, Levico Terme, 20-25 June 2016
Triangulated categories and geometry - a conference in honour of Amnon Neeman, Bielefeld, 15-19 May 2017

Papers
Adjoints to a Fourier-Mukai transform
In this paper I compute explicit formulas for the adjoints to a Fourier-Mukai functor, in a high level of generality. Namely, I work with quasi-compact schemes X and Y essentially of finite type over a Noetherian scheme. I develop the theory in two settings, either imposing a finite Tor dimensionality condition for Y , or asking that Y be quasi projective for the left adjoint (and the same conditions on X for the right adjoint). Instead of asking, as one traditionally does, that the kernel in D(XxY) be perfect, I impose the much weaker condition of perfection with respect to one of the projections to X and Y. It is surprising that such elegant formulas exist in this level of generality, and much of the power of this result is that functorial properties (like the two-out-of-three property) hold for adjoints in this much more general setting. This result has already found applications in Donovan-Wemyss's twist autoequivalences.

An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves (with M. Van den Bergh)
In this paper we construct the first example of an exact functor between the bounded derived categories of coherent sheaves of two projective varieties that is not isomorphic to a Fourier-Mukai functor. Specifically, the functor is from the derived category of a smooth quadric hypersurface in four-dimensional projective space to the derived category of P4. The question whether any such functor is Fourier-Mukai was open since Orlov's representability theorem in 1996, and previously there was no example of a non-Fourier- Mukai functor in the projective setting. This result falsifies a conjecture by Bondal, Larsen, and Lunts. The construction of the functor uses techniques from noncommutative deformation theory, A-infinity techniques, Massey products, as well as techniques inspired from algebraic topology such as Moore objects. Part of this construction was inspired by Neeman's paper on lifting stable homotopy in the category of spectra to a triangulated functor - one of the rare examples in the literature where one constructs a functor which does not descend from model categories.

Scalar extensions of derived categories and non-Fourier-Mukai functors (with M. Van den Bergh), Advances in Mathematics (2015), pp. 1100-1144
This paper continues to develop the scalar extensions of derived categories that I started in my paper below. Together with Van den Bergh, I developed the obstruction theory necessary to show that the functor in my paper below is not essentially surjective in general, and in particular it is not essentially surjective when B is a field of transcendence degree greater or equal to 3. This paper contains the first example of a non-Fourier-Mukai functor, but its target is the unbounded derived category of a variety as opposed to the bounded derived category. A lot more work needed to be done to obtain the example in the bounded case.

Representations of cohomological functors over extension fields, to appear in J. Noncomm. Geom.
In this paper I started the study of scalar extensions of derived categories, with the objective of applying it to the study of Fourier-Mukai functors. The idea is to consider the forgetful functor F:D(C_B) -> D(C)_B for C a k-linear Grothendieck category and B a k-algebra, where D(C)_B is the category of pairs (C,ρ) where C is an object of D(C) and ρ:B->Hom(C,C) is a k-algebra morphism. This innocuous-looking functor is harder to understand, and more interesting, than one might expect at first sight. In this paper I proved an essential surjectivity result for F when B is a field with transcendence degree over the base field less than or equal to 2, and the corresponding "generic representability" for an exact functor D(X) -> D(Y) when the dimension of Y is smaller or equal than 2. This suggested that to find examples of non-Fourier-Mukai functors one should be working in higher dimension.

On the existence of Fourer-Mukai kernels
In this paper I showed how to explicitly compute an approximation to a Fourier-Mukai functor of a given functor between derived categories of sheaves on two smooth projective varieties. Specifically, my method computes the virtual cohomology groups of the kernel. I believe that these will be fundamental in further understanding the behavior of any given functor. In particular, the results of this paper make it possible to give a partial explicit description of the non-Fourier-Mukai functor in my joint paper An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves.

Working Seminars
Spring 2014: I coordinated a joint SISSA/ICTP working seminar on Bridgeland Stability Conditions. Check out the videos of our lectures!
Fall 2012: I coordinated a graduate working seminar on Homological Projective Duality.
Spring 2012: I coordinated an undergraduate learning seminar on Percolation Theory.


Teaching
Calculus 2, Summer 2012
Calculus 2, Spring 2010
Calculus 1, Fall 2009
Calculus 1, Summer 2009