Images of Rational Maps of Projective Spaces.
July 14, 2019
Abstract: Consider a rational map from a projective space to a product of projective spaces, induced by a collection of linear projections. Motivated by the the theory of limit linear series and Abel-Jacobi maps, we study the basic properties of the closure of the image of the rational map using a combination of techniques of moduli functors and initial degenerations. We first give a formula of multi-degree in terms of the dimensions of intersections of linear subspaces and then prove that it is Cohen-Macaulay. Finally, we compute its Hilbert polynomials.
Keywords: projective spaces, moduli functors, multi-degrees, Chow rings, initial ideals, Hilbert polynomials.
Let be a vector space over an algebraically closed field , and consider the following rational map:
which is induced by the choice of linear subspaces . Let be the closure of the image of . The purpose of this paper is to study the basic properties of , using techniques of moduli functors and initial degenerations.
Fix a choice of linear subspaces with , and let be the dimension of and define to be the set:
In this paper, we have:
Theorem 1.1\normalshape. —
Set . The dimension of is . Its multi-degree function takes value one at the integer vectors in and zero otherwise. The Hilbert Polynomial of is
where ’s are the variables and is the smallest -th component of all elements of . Moreover, is Cohen-Macaulay.
Questions concerning some basic geometric properties of naturally arise in many settings, some of which are detailed below.
First, in [est-oss] by Esteves and Osserman, the closures of rational maps of the following form
arise as irreducible components of the closed subscheme of the fiber of the Abel map associated to a limit linear series. Moreover, the union of those irreducible components forms a flat degeneration of a projective space. In this work, they work with singular curves of compact type with two components, so they only need to study the closure of the image in a product of two projective spaces, where .
Secondly, in computer vision [computer] by Aholt, Sturmfels and Thomas, an important theme is to study the geometry of the closure of the rational map:
In their case, all the ’s are one dimensional and have pairwise trivial intersection.
Moreover, in the theory of Mustafin varieties in [mus] and [non-arc-unif], flat degenerations of are carefully studied, and the spaces arise as irreducible components of the Mustafin degenerations.
Finally, manifest themselves in algebraic statistics in [sta] by Morton .
Despite the ubiquity of the spaces , a systematic study of its the geometry (multi-degree, singularity type, initial degeneration, Hilbert Polynomial etc.) has not be conducted in full generality. Such generality is needed. The author’s original motivation is to generalize the work of Esteves and Osserman to arbitrary curves of compact type. In this case, one cannot just focus on the map to a product of two projective spaces, and also the vector subspaces ’s do not mutually just have trivial intersections anymore, but rather the only restriction we may impose is that . We don’t put restrictions on the dimension of and ’s. Moreover, we hope that our generality can lead to better understanding of irreducible components of Mustafin Degenerations.
In Section II, we will describe two closed subschemes and of , and prove that they are equal and that they agree with set-theoretically.
In Section III we will compute the multi-degree of . It seems to be an elementary linear algebra argument, but the proof involves moduli-theoretic techniques to exhibit the existence of the system of linear subspaces satisfying the desired property and such technique is also used to compute the deformation.
In the Section IV we will compute the initial degeneration of under a prescribed term order. The slogan is that multi-degrees determine the initial degenerations of . We also prove that , and are isomorphic as schemes, and finally conclude that is Cohen-Macaulay.
Section V will be devoted to computing the multi-variable Hilbert polynomials. Our computation of the Hilbert polynomial is not by counting monomials using the information of the initial ideal obtained in Section III, but rather by directly computing the Hilbert polynomial of its initial degeneration, which is a union of product of projective spaces whose intersections are also products of projective spaces, which is easier and more intuitive to deal with.
The author would like to thank his adviser Brian Osserman for his detailed and patient guidance. The author also wants to thank Professor Bernd Sturmfels for helpful discussions relating this work with [mus] and [sta], and also Professor Allen Knutson for introducing the author [mulfree] which reduces problem of “Cohen-Macaulay”ness to the proof of “multiplicity free”.
Also special thanks to Naizhen Zhang and Christopher Westenberger for encouraging my studies in algebraic geometry and Federico Castillo for explaining to me some foundational material in combinatorial commutative algebra and useful discussions. The author also wants to thank Michael Gröchenig for carefully proofreading this paper.
In this section, we first give a set theoretic description of . A scheme theoretic description of will be given in Section 2.3. Observing that -valued points of correspond to - tuples , where each is a linear subspace of which contains as codimension one linear subspace, we have the following theorem.
Proposition 2.1\normalshape. —
The closure of the image of the rational map are set-theoretically in bijection with -tuples where contains as a codimension one subspace and for any , we have .
Before proving the statement, let’s first start with an example:
Example 2.1\normalshape. —
Let be a five-dimensional linear subspace over a field with basis . Let , and . Let and .
We have , but , which means is not in the image of . We also have , and . Consider the following one-parameter family: . When , it is the point , when , except for possibly finite many values of . In this example, is not in the image of , but is in the closure of the image of
Proof of Proposition 2.1:
Note that -valued points in the image of the rational map correspond to where , thus one containment follows. For the other containment, given satisfying , for all , we will construct a one-parameter family , such that when , we have with except for possibly finite many values of , and when .
Let , then take a . Let . Then take a , and let , and take . Repeating this process, i.e. choose a , and let so that we have
Each is finite, thus for some we have . For , we can express (note that and are not necessarily the same).
Now consider the following set parameterized by :
It suffices to consider the one parameter family component wise: for component , we can find , such that . Let .
When , we have the point we start with. When , we have , thus is in the image of except for possibly finitely many values of . Then the proposition follows.
Given coordinate systems for all and ’s, let be the matrix representation of the linear map , and be . Set to be a subset of , and consider the following matrix associated to ,
Denote the ideal generated by all of the -minors of as , and denote the ideal generated by as . Description II of is the closed subscheme of cut by .
Definition 2.2\normalshape. —
Let be a finite index set and be a system of linear subspaces of , and with . Then define to be the following map:
induced by the following data:
the natural quotient map.
is the natural composed map , and is the zero map for .
Given any which contains as a codimension one subspace, corresponds to a rational point of through the natural map . Conversely, given a rational point of , which corresponds to a one-dimensional subspace of , take the preimage of under the natural quotient map , then one get a linear subspace of containing as a codimension one subspace. Given a homogeneous coordinate of , which is equivalent to giving a basis of , the matrix representation of is the same as the homogeneous coordinate of the point corresponding to up to a scalar. One can see that the matrix representation of is the same as .
One can immediately get that Description II agrees with at least set-theoretically:
Proposition 2.3\normalshape. —
Let be a vector space over a field with dimension . Let be a finite index set and be a system of linear subspaces of , and with . Then if and only if the following naturally induced map:
has , where is defined in Definition 2.2.
The condition is equivalent to the condition that has nontrivial kernel. The kernel of is . The kernel is nontrivial if and only if , which is equivalent to the condition that .
In this section I will prove that not just set-level, but also as schemes, the two descriptions in subsection Document and Document are the same. What we have done essentially in Proposition 2.3 is compare -valued points of the two descriptions. In this section, in order to compare the scheme structures of the two descriptions, we are going to compare their functor of points.
Before giving the definition of and , we first define a moduli functor , which contains and as closed sub-functors.
Definition 2.4\normalshape. —
Define to be a functor :
For any scheme , we associate the set of tuples sub-vector bundles of with , for any . And for each , the sub-bundle contains as a sub-bundle.
Note that the moduli functor represents nothing else than . The following two functors and represent closed subschemes of .
Definition 2.5\normalshape. —
Define to be a functor :
Given a scheme , we associate the set of tuples sub-vector bundles of with for any such that:
for each , the sub-bundle also contains as a sub-bundle
for any and the following bundle map:
In order the define the second functor, we need the following lemma:
Lemma 2.6\normalshape. —
Let be a vector space over and is a sub vector space of . For a scheme , consider the following exact sequence: