\begin{prop}[{cf.~\cite[Corollary~3.4]{S78}}]\label{prop:ci}
Let $R$ be a standard graded $\kk$-algebra.
If $R$ is a complete intersection, then the $h$-polynomial of $R$ is of type~CI, that is, $h_R(t)=\prod_{i=1}^r\Psi_{m_i}(t)$ for some non-negative integer $r$ and positive integers $m_1, \dots, m_r$.
\end{prop}

Throughout this paper, let $\Bbbk$ denote a field. We say that $R$ is a \emph{standard graded} $\Bbbk$-algebra
if $R = \bigoplus_{n \ge 0} R_n$ is a commutative graded $\Bbbk$-algebra with $R_0 = \Bbbk$ and finitely generated in degree one.
(A standard graded $\Bbbk$-algebra is also called a \emph{homogeneous} $\Bbbk$-algebra in other literature.)

\subsection{Background}
The Hilbert series of a (not necessarily commutative) graded $\Bbbk$-algebra is one of the most fundamental invariants in the study of graded $\Bbbk$-algebras.
It provides a concise analytic expression encoding the dimensions of the homogeneous components and thus captures the growth and structural features of the $\Bbbk$-algebra as a $\Bbbk$-vector space.
It is a classical and well-known fact that if a standard graded $\Bbbk$-algebra is Cohen-Macaulay, then the numerator polynomial of its Hilbert series, called the \textit{$h$-polynomial}, has all positive coefficients.
This fact is one of the cornerstones in graded commutative algebra.
Furthermore, several algebraic properties can be effectively characterized in terms of the Hilbert series.
A prominent theorem is Stanley's characterization of the Gorenstein property via the symmetry of the Hilbert series (\cite{S78}),
which has inspired subsequent studies establishing necessary conditions for generalized notions of Gorensteinness,
such as almost Gorensteinness (\cite{H}), nearly Gorensteinness (\cite{M}), and Artin-Schelter Gorensteinness (\cite{JZ}).

It is also known that when a standard graded $\Bbbk$-algebra is a complete intersection, its Hilbert series takes a particularly simple form.
In fact, for a positive integer $m$, let \[\Psi_m(t)=1+t+\cdots+t^{m-1}.\]
Then the $h$-polynomial of a standard graded complete intersection is of the form of a product of a finite collection of $\Psi_m(t)$'s (see Proposition~\ref{prop:ci}).
In addition, the Hilbert series also serves as an essential tool in investigating noncommutative graded complete intersections (see \cite{KKZ, KKZ2}).

\subsection{Cyclotomic standard graded $\Bbbk$-algebras and polynomials of type CI}

In recent years, increasing attention has been directed to the finer analytic behavior of the Hilbert series of standard graded $\Bbbk$-algebras, particularly the roots of their $h$-polynomials.
These roots encode subtle structural information and sometimes admit striking combinatorial interpretations
(see, e.g., the survey \cite{Br} for combinatorial consequences of the real-rootedness of the numerator polynomials of generating series).

Motivated by this perspective, we focus on the following notion for standard graded $\Bbbk$-algebras:

\begin{dfn}[cf. \cite{BD, KKZ}]
A standard graded $\Bbbk$-algebra $R$ is called \textit{cyclotomic} if all roots of its $h$-polynomial $h_R(t)$ lie on the unit circle in the complex plane.
\end{dfn}

Monic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \textit{Kronecker polynomials}.
It is known that every such polynomial can be expressed as a product of cyclotomic polynomials.
Note that the $h$-polynomials of cyclotomic $\Bbbk$-algebras are \textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials.
We say that a polynomial is \textit{of type CI} if it can be written as a product of $\Psi_m(t)$'s.
In particular, the $h$-polynomials of standard graded complete intersections are of type CI.
However, the converse does not hold in general; see, for example, \cite[Example~3.7]{S78}.
Borzi and D'Al\`i showed that the converse does hold when $R$ is Koszul~\cite[Theorem~3.9]{BD}.
Moreover, it was proved that a cyclotomic polynomial $\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\Bbbk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\cite[Theorem~4.3]{BD}.

Despite these results, in general, one cannot expect the ring-theoretic structure of $R$ to be determined by the cyclotomicity of its $h$-polynomial.
This naturally leads to the following question:
\begin{q}
Which polynomials can occur as the $h$-polynomials of cyclotomic standard graded $\Bbbk$-algebras, other than those of type CI?
\end{q}
For example, \cite[Theorem~4.3]{BD} indicates that a single cyclotomic polynomial $\Phi_m(t)$ is rarely equal to the $h$-polynomial of a standard graded $\Bbbk$-algebra.

\subsection{Main Results}

The aim of this paper is to make a contribution to this question.
Our main result is stated as follows.

\subsection{Background}
The Hilbert series of a (not necessarily commutative) graded $\Bbbk$-algebra is one of the most fundamental invariants in the study of graded $\Bbbk$-algebras.
It provides a concise analytic expression encoding the dimensions of the homogeneous components and thus captures the growth and structural features of the $\Bbbk$-algebra as a $\Bbbk$-vector space.
It is a classical and well-known fact that if a standard graded $\Bbbk$-algebra is Cohen-Macaulay, then the numerator polynomial of its Hilbert series, called the \textit{$h$-polynomial}, has all positive coefficients.
This fact is one of the cornerstones in graded commutative algebra.
Furthermore, several algebraic properties can be effectively characterized in terms of the Hilbert series.
A prominent theorem is Stanley's characterization of the Gorenstein property via the symmetry of the Hilbert series (\cite{S78}),
which has inspired subsequent studies establishing necessary conditions for generalized notions of Gorensteinness,
such as almost Gorensteinness (\cite{H}), nearly Gorensteinness (\cite{M}), and Artin-Schelter Gorensteinness (\cite{JZ}).

\begin{dfn}[cf. \cite{BD, KKZ}]
A standard graded $\Bbbk$-algebra $R$ is called \textit{cyclotomic} if all roots of its $h$-polynomial $h_R(t)$ lie on the unit circle in the complex plane.
\end{dfn}

Monic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \textit{Kronecker polynomials}.
It is known that every such polynomial can be expressed as a product of cyclotomic polynomials.
Note that the $h$-polynomials of cyclotomic $\Bbbk$-algebras are \textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials.
We say that a polynomial is \textit{of type CI} if it can be written as a product of $\Psi_m(t)$'s.
In particular, the $h$-polynomials of standard graded complete intersections are of type CI.
However, the converse does not hold in general; see, for example, \cite[Example~3.7]{S78}.
Borzi and D'Al\`i showed that the converse does hold when $R$ is Koszul~\cite[Theorem~3.9]{BD}.
Moreover, it was proved that a cyclotomic polynomial $\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\Bbbk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\cite[Theorem~4.3]{BD}.

Despite these results, in general, one cannot expect the ring-theoretic structure of $R$ to be determined by the cyclotomicity of its $h$-polynomial.
This naturally leads to the following question:
\begin{q}
Which polynomials can occur as the $h$-polynomials of cyclotomic standard graded $\Bbbk$-algebras, other than those of type CI?
\end{q}
For example, \cite[Theorem~4.3]{BD} indicates that a single cyclotomic polynomial $\Phi_m(t)$ is rarely equal to the $h$-polynomial of a standard graded $\Bbbk$-algebra.

\subsection{Main Results}

The aim of this paper is to make a contribution to this question.
Our main result is stated as follows.

\begin{abstract}
We call a standard graded commutative $\kk$-algebra \textit{cyclotomic} if its $h$-polynomial has all its roots on the unit circle in the complex plane.
Complete intersections provide typical examples of cyclotomic algebras,
since the $h$-polynomial of any standard graded complete intersection is a product of polynomials of the form $1 + t + \cdots + t^{m-1}$.
We refer to such polynomials as being \textit{of type CI}.
A natural question is whether there exists a cyclotomic standard graded $\kk$-algebra whose $h$-polynomial is not of type CI.
In this paper, we give a partial answer to this question.
We show that the $h$-polynomial $h_R(t)$ of a cyclotomic standard graded $\kk$-algebra $R$ is of type CI whenever $h_R(1) \in \{1, 4, 6\}$ or $h_R(1)$ is prime.
On the other hand, if $n \ge 8$ and $n$ is not prime, then there exists a cyclotomic standard graded $\kk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1) = n$.
\end{abstract}

Monic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \textit{Kronecker polynomials}.
It is known that every such polynomial can be expressed as a product of cyclotomic polynomials.
Note that the $h$-polynomials of cyclotomic $\kk$-algebras are \textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials.
We say that a polynomial is \textit{of type CI} if it can be written as a product of $\Psi_m(t)$'s.
In particular, the $h$-polynomials of standard graded complete intersections are of type CI.
However, the converse does not hold in general; see, for example, \cite[Example~3.7]{S78}.
Borzi and D'Al\`i showed that the converse does hold when $R$ is Koszul~\cite[Theorem~3.9]{BD}.
Moreover, it was proved that a cyclotomic polynomial $\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\kk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\cite[Theorem~4.3]{BD}.

The aim of this paper is to make a contribution to this question.
Our main result is stated as follows.

Note that $h_R(1)$ is nothing but the multiplicity of $R$ (see \cite[Proposition 4.1.9]{BH}), which is one of the important invariants of graded modules.

\section{Proof of Theorem~\ref{thm:main} (2)}\label{sec:(2)}
The goal of this section is to give a proof of Theorem~\ref{thm:main} (2).
Specifically, for any non-prime $n$ with $n \geq 8$, we construct a cyclotomic standard graded $\kk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.
Such $\kk$-algebras are realized as Stanley-Reisner rings of certain simplicial complexes.

We prove the following theorem, which implies Theorem~\ref{thm:main} (2).
\begin{thm}\label{thm:p=3}
{\em (1)} Let $q \ge 5$ be an odd integer.
Then there exists a cyclotomic standard graded $\kk$-algebra $R$ such that $h_R(t)$ is not of type CI with $h_R(1)=2q$.

\noindent
{\em (3)} Let $5 \le p \le q$ be odd integers.
Then there exists a cyclotomic standard graded $\kk$-algebra $R$ such that $h_R(t)$ is not of type CI with $h_R(1)=pq$.
\end{thm}

We postpone the proof of this theorem. So far, assume Theorem~\ref{thm:p=3} holds. Then we can prove the desired result. In fact,
let $R$ be a cyclotomic standard graded $\kk$-algebra whose $h$-polynomial is not of type CI.
Then, for any positive integer $a$, the standard graded $\kk$-algebra $R'=R[x]/(x^a)$ has the $h$-polynomial $h_{R'}(t)$ which is not of type CI and $h_{R'}(1)=a\cdot h_R(1)$.
(In fact, we have $h_{R'}(t)=h_R(t)\Psi_a(t)$.)

\begin{thm}[Main Result]\label{thm:main}
{\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\kk$-algebra $R$ is of type CI if $h_R(1) \in \{1,4,6\}$ or $h_R(1)$ is prime.

\noindent
{\em (2)} There exists a cyclotomic standard graded $\kk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$.
\end{thm}

Note that $h_R(1)$ is nothing but the multiplicity of $R$ (see \cite[Proposition 4.1.9]{BH}), which is one of the important invariants of graded modules.

\subsection{Structure of the paper}
We briefly describe the structure of this paper.
In Section~\ref{sec:pre}, we fix our notation (Hilbert series, $h$-polynomials, simplicial complexes, etc.),
review fundamental notions (cyclotomic polynomials, Stanley-Reisner theory, and related topics),
and recall classical results, namely Macaulay's theorem and the $g$-theorem, which will be used in the proof of Theorem~\ref{thm:main}.
Sections~\ref{sec:(1)} and~\ref{sec:(2)} are devoted to the proofs of parts~(1) and~(2) of Theorem~\ref{thm:main}, respectively.

The post-theorem introduction gives only a high-level roadmap: it says that Section~\ref{sec:pre} recalls classical results—"namely Macaulay's theorem and the $g$-theorem"—"which will be used in the proof of Theorem~\ref{thm:main}," and that Sections~\ref{sec:(1)} and~\ref{sec:(2)} contain the proofs of parts~(1) and~(2) of Theorem~\ref{thm:main}, respectively.

The post-theorem introduction gives only a high-level roadmap: it says that next it recalls classical results—"namely Macaulay's theorem and the $g$-theorem"—"which will be used in establishing the main theorem," and that later it contains the proofs of parts~(1) and~(2) of the main theorem, respectively.

[Main Result]\label{thm:main}
{\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\Bbbk$-algebra $R$ is of type CI if $h_R(1) \in \{1,4,6\}$ or $h_R(1)$ is prime.

\noindent
{\em (2)} There exists a cyclotomic standard graded $\Bbbk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$.,

Question: Let $R$ be a standard graded commutative $\Bbbk$-algebra with Hilbert series of the form $H_R(t)=\dfrac{h_R(t)}{(1-t)^d}$, where $h_R(t)$ is its $h$-polynomial. Call $R$ cyclotomic if all roots of $h_R(t)$ lie on the unit circle in $\mathbb{C}$. A polynomial is said to be of type CI if it can be written as a product of polynomials $\Psi_m(t)=1+t+\cdots+t^{m-1}$. Which statement holds for such algebras and $h$-polynomials?

For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, for every non-prime integer $n\ge 8$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.

Choices:
B. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6,8\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, for every non-prime integer $n\ge 9$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.
C. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI.
D. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, there exists a non-prime integer $N\ge 8$ such that for every non-prime integer $n\ge N$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ with $h_R(t)$ not of type CI and $h_R(1)=n$.
E. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is composite, then $h_R(t)$ is of type CI. Moreover, for every prime integer $n\ge 8$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.

sharp exceptional values and threshold $n\ge 8$

dropped the existence clause in part (2)

uniform bound replaced exact range of all non-prime $n\ge 8$

prime/non-prime dichotomy in both directions


\label{thm:deg}
Consider a projective variety $X \subset {\mathbb P}^n$ (over an algebraically closed field ${\mathbb K}$ of characteristic zero),
together with a (reduced) simple normal crossings (snc) divisor $E \subset {\mathbb P}^n$.
Assume that the degrees of both $X$ and $E$ are at most $d$.
Then there is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities
$(X',E')$, where $(X',E')$ can be embedded in ${\mathbb P}^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in ${\mathbb P}^{n'}$.

\begin{remark}[\bf Summary remark]\label{rem:summaryremark}
The data involved in resolution of singularities of a marked ideal $\ucI = (Z,X,E,\cI,\mu)$
with initial data $\ga = (0,n,m,d,l,q,\mu)$, have bounds given by the recursive functions above, with final data
vector majorized by $\Ga^{(m)}(\ga)$.

As in \cite{BGMW}, we can also keep track of the Grzegorczyk complexity classes of the resolution data.
Beginning with Step II above, $M(n,d) \in \cE^3$ (cf. Lemma \ref{lem:mult}; we recall that, in general, $\cE^1$ contains
all linear functions, $\cE^2$ all polynomials, and $\cE^3$ all towers of exponential functions$,\ldots$). In Step IB, then $\De_1(\ga)
\in \cE^3$ and, by induction on $m$, $\Ga^{(m-1)}(\De_\mathrm{I}(\ga)) \in \cE^{m+2}$, and $\Ga_{\mathrm{I}}^{(m)}(\ga) \in \cE^{m+2}$.
In Step IIA, $\Ga_{\mathrm{IIB}}^{(m)}(\ga) \in \cE^{m+3}$, from Lemmas \ref{lem:blupsummary} and \ref{lem:mult},
and finally $\Ga^{(m)}(\ga) \in \cE^{m+3}$,
in Step IIB. For more details, see \cite{BGMW}.
\end{remark}

\begin{theorem}\label{thm:deg}
Consider a projective variety $X \subset \IP^n$ (over an algebraically closed field $\IK$ of characteristic zero),
together with a (reduced) simple normal crossings (snc) divisor $E \subset \IP^n$.
Assume that the degrees of both $X$ and $E$ are at most $d$.
Then there is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities
$(X',E')$, where $(X',E')$ can be embedded in $\IP^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in $\IP^{n'}$.
\end{theorem}

\label{sec:intro}

The \emph{degree} $\deg X$ of an irreducible projective variety $X \subset {\mathbb P}^n$ (over an algebraically closed field ${\mathbb K}$)
denotes the number of intersection points of $X$ with a generic linear subspace $L$ of complementary dimension ($\dim L =
n - \dim X$). In general, the \emph{degree} of $X \subset {\mathbb P}^n$ is the sum of the degrees of the irreducible components of $X$
of all dimensions. The \emph{degree} of a divisor $E \subset {\mathbb P}^n$ likewise means the sum of the degrees of its (reduced) components (which
are not necessarily irreducible).

For example, if $X$ is a hypersurface and $F$ generates the ideal of homogeneous polynomials vanishing on $X$, then
$\deg X$ equals the degree of the homogeneous polynomial $F$.

The purpose of this article is to give a proof of the following result that Caucher Birkar posed to us as a question.

\begin{abstract}
Consider a projective variety $X \subset \IP^n$ (over an algebraically closed field of characteristic zero),
together with a (reduced) simple normal crossings divisor $E \subset \IP^n$, where
the degrees of both $X$ and $E$ are at most $d$. We show
there is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities
$(X',E')$, where $(X',E')$ can be embedded in $\IP^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in $\IP^{n'}$.
\end{abstract}

The \emph{degree} $\deg X$ of an irreducible projective variety $X \subset \IP^n$ (over an algebraically closed field $\IK$)
denotes the number of intersection points of $X$ with a generic linear subspace $L$ of complementary dimension ($\dim L =
n - \dim X$). In general, the \emph{degree} of $X \subset \IP^n$ is the sum of the degrees of the irreducible components of $X$
of all dimensions. The \emph{degree} of a divisor $E \subset \IP^n$ likewise means the sum of the degrees of its (reduced) components (which
are not necessarily irreducible).

The purpose of this article is to give a proof of the following result that Caucher Birkar posed to us as a question.

An snc divisor $E$ transforms by a blowing-up $\s$ with smooth centre $C$ that is snc with $E$, to an snc divisor defined
by the strict (or birational) transform of $E$ plus the exceptional divisor of $\s$.
\emph{Log resolution} of $(X,E)$ means a resolution of singularities $X' \to X$ given by a composite of smooth blowings-up
as above, such that $X', E'$ have only simple normal crossings, where $E'$ is the final transform of $E$.
If we assume that $E$ is ordered, then $E'$ is also ordered following the sequence of blowings-up.

\begin{theorem}\label{thm:gendeg}
Consider a projective variety $X \subset \IP^n$ (over an algebraically closed field $\IK$ of characteristic zero),
together with an snc divisor $E \subset \IP^n$.
We assume that $X$ and $E$
can be defined by homogeneous polynomials of degrees at most $d$. Then there is a pair
$(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities
$(X',E')$, where $(X',E')$ can be embedded in $\IP^{n'}$ and defined in the latter by homogeneous polynomials
of degrees at most $d'$.
\end{theorem}

Theorem \ref{thm:gendeg} is a consequence of estimates given in \cite{BGMW} or in Sections \ref{sec:affmarkedideal}--\ref{sec:summaryest}
below, in the context of log resolution of
singularities of an embedded algebraic variety $X \hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E \subset Z$.
The triple $(Z,X,E)$ can be defined locally by polynomial data in affine spaces $\IA^n$ with rational transition mappings,
and there is a log resolution of singularities $(X',E') \subset Z'$ which can be defined by data with effective bounds
on local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up
needed, etc., in terms of bounds on local data needed to define $(X,E) \subset Z$.
Example \ref{ex:proj} below shows how to apply these effective bounds to obtain Theorem \ref{thm:gendeg}.

\begin{example}\label{ex:proj} \emph{Case of a projective variety and snc divisor $(X,E) \subset \IP^n$.} We begin with
a marked ideal $\cI$ as in Example \ref{ex:princ,desing}(2), with $Z=\IP^n$, and we can reduce to the case
that $E=\emptyset$ by replacing $(X,E)$ by $(X\cup E,\,\emptyset)$. Then there is an \emph{affine marked ideal} $\ucT$
(a certain collection of resolution data associated to $\ucI$) as in
Section \ref{sec:affmarkedideal} below, where the affine charts
$(\IK^{n_\al})_\al$ are affine charts of $\IP^n$ (each $n_\al = n$). See Definition \ref{def:affmarkedideal} and Remark
\ref{rem:affmarkedideal}. In the initial data given by Definition \ref{notn:affmarkedideal}, all terms are bounded in a
simple way in terms of the given degree bounds of Theorem \ref{thm:gendeg}.

\begin{definition}\label{def:affmarkedideal}
An \emph{affine marked ideal} $\ucT$ is a collection of tuples together with an associated order $\mu$,
\begin{equation}\label{eq:affmarkedideal}
\ucT = \left(\left\{U_{\al\be}, X_{\al\be}, E_{\al\be}, \cI_{\al\be}, \left(\IK^{n_\al}\right)_\al: \al \in A,\,\be\in B_\al\right\}, \mu\right),
\end{equation}
where $A$ and the $B_\al$ are finite index sets, and
\begin{enumerate}
\item $(\IK^{n_\al})_\al \cong \IK^{n_\al}$, with affine coordinates $x_\al = (x_{\al 1},\ldots,x_{\al,n_\al})$;
\item $\{U_{\al\be}: \be\in B_\al\}$ is an open covering of $(\IK^{n_\al})_\al$, where $U_{\al\be} \subset (\IK^{n_\al})_\al$
is the complement of the zero set of a polynomial $f_{\al\be} \in \IK[x_\al]$;
\item $E_{\al\be}$ is a collection of smooth divisors in $(\IK^{n_\al})_\al$, each given by an equation $x_{\al j} = 0$,
for some $j=1,\ldots,n_\al$;
\item $X_{\al\be} \subset (\IK^{n_\al})_\al$ is a closed subset, where $X_{\al\be} \cap U_{\al\be}$ is smooth; moreover,
there is a set of parameters (coordinates) on $U_{\al\be}$,
$$
u_{\al\be, 1},\ldots,u_{\al\be,n_\al} \in \IK[x_\al],
$$
where each $u_{\al\be, i}$ is either a coordinate $x_{\al j}$ describing an exceptional divisor (i.e., an element of $E_{\al\be}$)
or is transverse to $E_{\al\be}$ over $U_{\al\be}$,
and $\cI_{X_{\al\be}}$ is the ideal
$$
\cI_{X_{\al\be}} = (u_{\al\be, 1},\ldots,u_{\al\be,n_\al-m}) \subset \IK[x_\al],
$$
with $u_{\al\be, 1},\ldots,u_{\al\be,n_\al-m}$ all transverse to $E_{\al\be}$;
\item $\cI_{\al\be}$ is an ideal $(g_{\al\be,1},\ldots,g_{\al\be,\overline{j}}) \subset \IK[x_\al]$,
and
$$
\cosupp (\cI_{\al\be},\mu) \cap U_{\al\be} \cap U_{\al\be'} = \cosupp (\cI_{\al\be'},\mu) \cap U_{\al\be} \cap U_{\al\be'},
$$
where $(\cI_{\al\be},\mu)$ denotes the marked ideal $(U_{\al\be},X_{\al\be}\cap U_{\al\be}, E_{\al\be}, \cI_{\al\be},\mu)$;
\item for all $\al_1,\al_2 \in A$, $\be_1\in B_{\al_1}$, $\be_2\in B_{\al_2}$,
there exist
\begin{align*}
v_{\al_1\be_1\al_2\be_2,1},\ldots,v_{\al_1\be_1\al_2\be_2,n_{\al_2}} &\in \IK[x_{\al_1}],\\
w_{\al_1\be_1\al_2\be_2,1},\ldots,w_{\al_1\be_1\al_2\be_2,n_{\al_2}} &\in \IK[x_{\al_1}],
\end{align*}
such that
$$
x_{\al_1} \mapsto \left(\frac{v_{\al_1\be_1\al_2\be_2,1}}{w_{\al_1\be_1\al_2\be_2,1}},\ldots,\frac{v_{\al_1\be_1\al_2\be_2,n_{\al_2}}}{w_{\al_1\be_1\al_2\be_2,n_{\al_2}}}\right)(x_{\al_1})
$$
induces a birational mapping $i_{\al_1\be_1\al_2\be_2}: X_{\al_1\be_1} \dashrightarrow X_{\al_2\be_2}$;
moreover, the birational mappings $i_{\al_1\be_1\al_2\be_2}$ determine a variety $X_{\ucT}$, unique up to isomorphism,
together with open embeddings $j_{\al\be}: X_{\al\be} \cap U_{\al\be} \hookrightarrow X_{\ucT}$ defining an open covering
of $X_{\ucT}$, such that $j_{\al_2\be_2}^{-1}\circ j_{\al_1\be_1} = i_{\al_1\be_1\al_2\be_2}$;
\item $\mu$ is a nonnegative integer.
\end{enumerate}
\end{definition}

An snc divisor $E$ transforms by a blowing-up ${\sigma}$ with smooth centre $C$ that is snc with $E$, to an snc divisor defined
by the strict (or birational) transform of $E$ plus the exceptional divisor of ${\sigma}$.
\emph{Log resolution} of $(X,E)$ means a resolution of singularities $X' \to X$ given by a composite of smooth blowings-up
as above, such that $X', E'$ have only simple normal crossings, where $E'$ is the final transform of $E$.
If we assume that $E$ is ordered, then $E'$ is also ordered following the sequence of blowings-up.

We prove, in fact, the following variant of Theorem \ref{thm:deg}; the two assertions are equivalent because of the degree
bounds in Section \ref{sec:deg} below

\begin{theorem}\label{thm:gendeg}
Consider a projective variety $X \subset {\mathbb P}^n$ (over an algebraically closed field ${\mathbb K}$ of characteristic zero),
together with an snc divisor $E \subset {\mathbb P}^n$.
We assume that $X$ and $E$
can be defined by homogeneous polynomials of degrees at most $d$. Then there is a pair
$(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities
$(X',E')$, where $(X',E')$ can be embedded in ${\mathbb P}^{n'}$ and defined in the latter by homogeneous polynomials
of degrees at most $d'$.
\end{theorem}

Theorem \ref{thm:gendeg} is a consequence of estimates given in \cite{BGMW} or in Sections \ref{sec:affmarkedideal}--\ref{sec:summaryest}
below, in the context of log resolution of
singularities of an embedded algebraic variety $X \hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E \subset Z$.
The triple $(Z,X,E)$ can be defined locally by polynomial data in affine spaces ${\mathbb A}^n$ with rational transition mappings,
and there is a log resolution of singularities $(X',E') \subset Z'$ which can be defined by data with effective bounds
on local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up
needed, etc., in terms of bounds on local data needed to define $(X,E) \subset Z$.
Example \ref{ex:proj} below shows how to apply these effective bounds to obtain Theorem \ref{thm:gendeg}.

The estimates in the article show that the contribution of the dimension $n$ to $(n',d')$ in the theorems above
dwarfs that of the degree $d$. This is highlighted in \cite{BGMW} by a complexity bound in terms of
Grzegorczyk complexity classes ${\mathcal E}^l$, $l\geq 0$, of primitive recursive (integer) functions,
where the functions in each ${\mathcal E}^l$ require at most $l$ nested primitive recursions \cite{Grz}, \cite{WW}. The number of nested
recursions involved in the desingularization algorithm for $(X,E) \subset {\mathbb P}^n$, is bounded by $n + 3$ (cf.
Remark \ref{rem:summaryremark}).

We use the algorithm for functorial resolution of singularities as presented in \cite{BMinv}, \cite{BMfunct} (the version
in \cite{Wlodar} was used in \cite{BGMW}). Log resolution of singularities
of an embedded pair $(X,E) \subset Z$, or of an ideal ${\mathcal I} \subset \cO_Z$ together with an snc divisor $E\subset Z$,
follows from resolution of singularities of a collection of resolution data called a \emph{marked ideal}. (Non-embedded)
desingularization of a pair $(X,E)$ follows from functoriality of the embedded case (applied locally).

There are several articles in the literature on implementation of algorithms for resolution of singularities (e.g., \cite{BS},
\cite{F-KP}). Sections \ref{sec:affmarkedideal}-\ref{sec:bounds} below can be compared with explicit computation of marked ideals
in \cite{BS}, which raises the challenge of ``super-exponential growth of exponents'' for generators of coefficient marked ideals.
The purpose of our methods is to provide effective bounds on measures of the complexity of the algorithm.

The post-theorem text does not give step-by-step “main steps” for proving Theorem~\ref{thm:deg}, but it does indicate the proof strategy via a variant and where the needed bounds come from: one proves the variant Theorem~\ref{thm:gendeg}, and “the two assertions are equivalent because of the degree bounds in Section~\ref{sec:deg} below.” Then “Theorem~\ref{thm:gendeg} is a consequence of estimates given in \cite{BGMW} or in Sections~\ref{sec:affmarkedideal}--\ref{sec:summaryest} below,” namely effective bounds “in the context of log resolution of singularities of an embedded algebraic variety $X\hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E\subset Z$.” The argument is described as: define $(Z,X,E)$ locally by “polynomial data in affine spaces ${\mathbb A}^n$ with rational transition mappings,” apply the functorial resolution algorithm (“as presented in \cite{BMinv}, \cite{BMfunct}”) and the reduction to “a collection of resolution data called a \emph{marked ideal},” obtaining a log resolution $(X',E')\subset Z'$ “defined by data with effective bounds on local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up needed, etc., in terms of bounds on local data needed to define $(X,E)\subset Z$.” Finally, “Example~\ref{ex:proj} below shows how to apply these effective bounds to obtain Theorem~\ref{thm:gendeg},” and the non-embedded case follows since “(Non-embedded) desingularization of a pair $(X,E)$ follows from functoriality of the embedded case (applied locally).”

The post-theorem text does not give step-by-step “main steps” for proving the main theorem, but it does indicate the proof strategy via a variant and where the needed bounds come from: one proves the variant
\begin{theorem}\label{thm:gendeg}
Consider a projective variety $X \subset \IP^n$ (over an algebraically closed field $\IK$ of characteristic zero),
together with an snc divisor $E \subset \IP^n$.
We assume that $X$ and $E$
can be defined by homogeneous polynomials of degrees at most $d$. Then there is a pair
$(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities
$(X',E')$, where $(X',E')$ can be embedded in $\IP^{n'}$ and defined in the latter by homogeneous polynomials
of degrees at most $d'$.
\end{theorem}
(and “the two assertions are equivalent because of the degree bounds in Section~\ref{sec:deg} below.”). Then “Theorem~\ref{thm:gendeg} is a consequence of estimates given in \cite{BGMW} or in Sections~\ref{sec:affmarkedideal}--\ref{sec:summaryest} below,” namely effective bounds “in the context of log resolution of singularities of an embedded algebraic variety $X\hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E\subset Z$.” The argument is described as: define $(Z,X,E)$ locally by “polynomial data in affine spaces ${\mathbb A}^n$ with rational transition mappings,” apply the functorial resolution algorithm (“as presented in \cite{BMinv}, \cite{BMfunct}”) and the reduction to “a collection of resolution data called a \emph{marked ideal},” obtaining a log resolution $(X',E')\subset Z'$ “defined by data with effective bounds on local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up needed, etc., in terms of bounds on local data needed to define $(X,E)\subset Z$.” Finally, the following example shows how to apply these effective bounds to obtain Theorem~\ref{thm:gendeg}:

\begin{example}\label{ex:proj} \emph{Case of a projective variety and snc divisor $(X,E) \subset \IP^n$.} We begin with
a marked ideal $\cI$ as in Example \ref{ex:princ,desing}(2), with $Z=\IP^n$, and we can reduce to the case
that $E=\emptyset$ by replacing $(X,E)$ by $(X\cup E,\,\emptyset)$. Then there is an \emph{affine marked ideal} $\ucT$
(a certain collection of resolution data associated to $\ucI$) as in
Section \ref{sec:affmarkedideal} below, where the affine charts
$(\IK^{n_\al})_\al$ are affine charts of $\IP^n$ (each $n_\al = n$). See Definition \ref{def:affmarkedideal} and Remark
\ref{rem:affmarkedideal}. In the initial data given by Definition \ref{notn:affmarkedideal}, all terms are bounded in a
simple way in terms of the given degree bounds of Theorem \ref{thm:gendeg}.

If $\s: Z' \to Z$ is a blowing-up of $Z$ with smooth closed centre $C$, then $Z'$ is embedded in $\IP^n\times\IP^r$,
$r = \codim C - 1$, and the affine marked ideal $\ucT'$ defined over $\ucT$ by the blowing-up (Section \ref{sec:affmarkedideal})
has the affine charts of $\IP^n\times\IP^r$; $\ucT'$ corresponds to a marked ideal on the strict transform $Z'$ of $Z$
in $\IP^n\times\IP^r$.

The final affine marked ideal $\ucT_*$ given by the resolution algorithm corresponds to a marked ideal $\ucI_*$
on the final strict transform of $Z$ in a finite product $\IP^n\times\IP^{r_1}\times\IP^{r_2}\times\cdots$, where the codimensions
of the successive centres of blowing up are $r_1+1,\, r_2+1,\ldots$ in the successive products of projective spaces.
We can embed the final product in $\IP^{n'}$, for suitable $n'$, using the Segre embedding, and effective estimates
for $(X',E')$ follow from those of Section \ref{sec:summaryest}.
\end{example}

and the non-embedded case follows since “(Non-embedded) desingularization of a pair $(X,E)$ follows from functoriality of the embedded case (applied locally).”

Question: Let $X \subset \mathbb{P}^n$ be a projective variety over an algebraically closed field $\mathbb{K}$ of characteristic $0$, and let $E \subset \mathbb{P}^n$ be a reduced simple normal crossings (snc) divisor. For an irreducible projective variety, its degree is the number of intersection points with a generic linear subspace of complementary dimension; for a general projective variety or divisor, the degree is the sum of the degrees of its reduced irreducible components. A log resolution of $(X,E)$ means a resolution $X' \to X$ obtained by a composite of smooth blowings-up with smooth centers that have snc with the current transform of $E$, such that the resulting $X'$ and the final transform $E'$ have only simple normal crossings. Assume that $\deg X \le d$ and $\deg E \le d$. Which existence statement holds?

There exist integers $n'$ and $d'$, explicitly computable in terms of $(n,d)$, and a log resolution of singularities $(X',E')$ of $(X,E)$ such that $(X',E')$ can be embedded in $\mathbb{P}^{n'}$, with both $X'$ and $E'$ having degree at most $d'$ in $\mathbb{P}^{n'}$.

Choices:
B. There exist integers $n'$ and $d'$, explicitly computable in terms of $(n,d)$, such that for every such pair $(X,E)$ there is a log resolution of singularities $(X',E')$ of $(X,E)$ with $(X',E')$ embedded in $\mathbb{P}^{n'}$, where $X'$ has degree at most $d'$ in $\mathbb{P}^{n'}$ and $E'$ is a reduced snc divisor in $\mathbb{P}^{n'}$.
C. There exist integers $n'$ and $d'$, explicitly computable in terms of $(n,d)$, such that for every such pair $(X,E)$ there is a log resolution of singularities $(X',E')$ of $(X,E)$ with $(X',E')$ embedded in $\mathbb{P}^{n'}$.
D. For every such pair $(X,E)$ there exists a log resolution of singularities $(X',E')$ of $(X,E)$ and integers $n'$ and $d'$ such that $(X',E')$ can be embedded in $\mathbb{P}^{n'}$, with both $X'$ and $E'$ having degree at most $d'$ in $\mathbb{P}^{n'}$, where $n'$ and $d'$ depend only on $n$.
E. There exist integers $n'$ and $d'$, explicitly computable in terms of $(n,d)$, such that for every such pair $(X,E)$ one can choose a log resolution of singularities $(X',E')$ of $(X,E)$ with $(X',E')$ embedded in $\mathbb{P}^{n'}$, and such that $X'$ and $E'$ are each defined in $\mathbb{P}^{n'}$ by homogeneous polynomials of degrees at most $d'$.

dropped the degree bounds on $X'$ and $E'$ in $\mathbb{P}^{n'}$

dependence of $n',d'$ on $d$ as well as $n$

replaced bounded degree of the embedded subvarieties/divisors by bounded degrees of defining equations for the final pair


\begin{theorem}\label{thm_main}
Let $M$ be a closed topological manifold of dimension $2n$ and let $b = \dim H^2(M,\mathbb{C})$.

Then there exist non-zero elements $e_1,\ldots,e_q \in H^2(M,\mathbb{C})$ with $q \leq b+1$ such that if $X$ is a smooth complex projective variety of dimension $n$ with underlying topological space $M$ and $f\colon X \to Y$ is a divisorial contraction to a point with exceptional divisor $E$, then there exists $i \in \{1, \dots, q\}$ such that $[E] = e_i$.
\end{theorem}

This result provides a topological bound on the possible exceptional divisors arising from some of the steps of the Minimal Model Program, establishing a new link between topology and birational geometry. It is a consequence of Theorem \ref{thm:W_F}, where we prove the finiteness of rank one points of $\mathcal H_F$ where $F$ is a non-degenerate $n$-form, a result of independent interest.

\medskip

The structure of the paper is as follows. In Section \ref{s_preliminaries}, we introduce the necessary background on tensors and forms. Sections \ref{s_low} and \ref{s:intersection} are devoted to the study of the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$ with $n\ge 3$. We show, in particular, that, up to a non-zero scalar, there are at most finitely many  classes $e\in H^2(M,\mathbb C)$ such that
$e^n\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.
(cf. Theorem \ref{thm:W_F}). Section \ref{s_blowups} establishes our main theorem by examining the rank  of the associated  form in the presence of a divisorial contraction. In Section \ref{s_volume},  we present some related open problems.

\medskip

\textbf{Acknowledgements:}  The first and third authors are members of the GNSAGA - Istituto Nazionale di Alta Matematica.
The first author was also partially supported by the PRIN ``Vartietà reali e complesse: geometria, topologia e analisi armonica".
The second author is partially supported by a Simons collaboration grant  and would like to thank the National Center for Theoretical Sciences in Taiwan and Professor Jungkai Chen for
their  hospitality, where some of the work for this paper was completed.
The third author is partially supported by the PRIN2020 research grant ``2020KKWT53”.
The authors would also like to thank the referee for careful reading the paper and for several useful comments.

\section{Preliminaries}\label{s_preliminaries}
We work over the field of complex numbers $\mathbb C$. A \emph{form} over $\mathbb C$ is a homogeneous polynomial $F\in \mathbb C[x_0,\ldots,x_b]$.

\subsection{Tensors} Let $a_1,\dots,a_n$ be positive integers. A \emph{tensor} of type $a_1\times \ldots \times a_n$ is a multilinear map
$$T\colon \mathbb C^{a_1}\times \ldots \times \mathbb C^{a_n}\to \mathbb C.$$
For any positive integer $a$, denote $[a]:=\{ 1,2,\ldots, a \}$. Then a tensor $T$ of type  $a_1\times \ldots \times a_n$ is determined uniquely by a function
$$\tilde T\colon [a_1]\times \cdots \times [a_n] \to \mathbb{C}.$$
We will refer to $\tilde T$ as the \emph{hypermatrix associated to $T$}.
Given positive integers
$a_j^{'} \le a_j$ for $1 \le j \le n,$ and strictly increasing functions $f_j\colon [a_j^{'}] \to [a_j]$,
we define a \emph{sub-tensor} $T'$ of type $a'_1\times\ldots\times a'_n$ so that if $\tilde T'$ is the hypermatrix associated to $T'$ then
$$\tilde T'=\tilde T(f_1,\dots,f_n)\colon [a'_1]\times \cdots \times [a'_n] \to \mathbb{C}.$$
Note that if $v_i\in \mathbb C^{a_i}$ for $i=1,\dots,n$, then there exists a natural tensor of type $a_1\times \ldots \times a_n$ defined by
$v_1\otimes \ldots \otimes v_n$.

\begin{definition}
A nonzero tensor $T$ of type  $a_1\times \ldots \times a_n$ has {\it rank one} if there are nonzero vectors $v_i \in \mathbb{C}^{a_i}$ such that $T=v_1 \otimes \cdots \otimes v_n $. We define the
{\it rank} of a nonzero tensor $T$ ($=\rk(T)$) to be the minimum positive integer $r$
such that there exist $r$ tensors $T_1, \cdots , T_r$ of rank one with

$$T = T_1 + \cdots + T_r .$$
\end{definition}

Note that the rank of any sub-tensor of $T$ cannot be larger than the rank of $T$.
\begin{definition}\label{d_cubic}
A \emph{cubic tensor} $T$ is a tensor of type $d^{\times n} =d \times \cdots \times d$. Let $A$ be the hypermatrix associated to $T$.
We denote by $\det A$ the \emph{hyperdeterminant} of  $A$ (see \cite[Chapter 14]{GKZ} for the definition and some of its main properties). In particular, $\det A$  is a polynomial  in the entries of $A$.

\end{definition}

The following theorem is useful to compute the rank of a tensor:
\begin{theorem}{\cite[Thm. 3.1.1.1]{Landsberg-book}} \label{thm:slices}
Let $A_1,\dots,A_n$ be $l\times m$ matrices and consider the  tensor $A=[A_1,\ldots,A_n]$ of type $l\times m\times n$.

Then $\rk(A)$ is equal to  the minimum number of rank one matrices needed to span a vector space containing $\langle A_1,\ldots, A_n \rangle$.
\end{theorem}

\begin{proof}
Let $r=\rk(A)$ and let $B_1,\ldots, B_r$ be rank one matrices such that
$$\langle A_0,\ldots, A_q\rangle\subset \langle B_1,\ldots, B_r\rangle,$$
as in Theorem \ref{thm:slices}.
Thus, for any $j=0,\ldots,q$, we may write
$$
A_j= \sum_{i=1}^r \lambda_{ij}B_i
$$
for $\lambda_{ij} \in \C$ for $i=1,\ldots,r$ and $j=0,\dots,q$. Since $A_1,\ldots, A_q$ are linearly independent, after possibly reordering the $B_i$'s, we may assume that the $q\times q$-matrix
$$L=(\lambda_{ij})_{i,j=1,\ldots,q}$$
 is of maximal rank $q$. Thus, the linear system
$$L \cdot X =(-\lambda_{10},-\lambda_{20},\ldots,-\lambda_{q0})^t$$
admits a solution $(\mu_1,\ldots,\mu_q)\in \mathbb C^q$.
It follows that
 $$A_0 + \sum_{i=1}^q \mu_i A_i=\sum_{j=q+1}^{r} s_j B_j$$ for some  $s_{q+1},\ldots,s_r \in \C$.
 Since the rank of $\sum_{j=q+1}^{r} s_j B_j$ is at most $r-q$, our assumption implies that $r-q \ge t$  and the claim follows.
\end{proof}

\section{Points of low rank}\label{s_low}

\begin{definition}
	Given a form $F \in \C[x_0,\ldots,x_b]$ of degree $n \ge 3$ we consider the tensor
of type $(b+1)^{\times (n-1)}$
		defined by the hypermatrix $\mathcal H_F$ of the $(n-1)$-th order derivatives, whose entries are linear forms in $x_0, \ldots, x_b$.
	We say that $F$ is \emph{honest} if for any non-zero element $v \in \mathbb C^{b+1}$, we have  $\mathcal H_F(v) \ne 0$.
    We say that $F$ is  \emph{non-degenerate} if $ \det \mathcal H_F(v) \ne 0$ for some
 non-zero $v \in \mathbb C^{b+1}$
 (note that this definition is consistent with \cite[Page 7929]{CT18} but not with \cite[14.1.A]{GKZ}).
\end{definition}

Note that the locus $p \in \mathbb P^b$ such that  $\mathcal H_F(p) = 0$ is well defined. By \cite[Remark 6.3.5]{Bocci-Chiantini}, the rank $\rk \mathcal H_F(p)$ is also well-defined.

\begin{example}\label{ex:degenerate}
	Consider the  form
	$$
	F(x_0,\ldots,x_4)=\frac{x_0x_1^2}{2} + x_1x_3x_4 + \frac{x_2x_3^2}{2}.
	$$
	Then

	$$
	\mathcal H_F=\begin{pmatrix}
	0   & x_1 & 0   & 0   & 0   \\
	x_1 & x_0 & 0   & x_4 & x_3 \\
	0   & 0   & 0   & x_3 & 0   \\
	0   & x_4 & x_3 & x_2 & x_1 \\
	0   & x_3 & 0   & x_1 & 0
	\end{pmatrix}
	$$
Thus, the form $F$ is honest, but it is degenerate since $\det \mathcal H_F$ is identically zero.
\end{example}

Vaguely speaking, an honest form can be characterised by the fact that it  depends on all the variables. More precisely, we have:
\begin{lemma}\label{lem:H_F(p)=0}
	Let $F \in \C[x_0,\ldots,x_b]$ be a form of degree $n$ and let
    $p:=[1,0,\ldots,0] \in \mathbb P^b$.

    Then $\mathcal H_F(p) = 0$ if and only if $x_0$ does not appear in the expression of $F$.  In particular, a non-degenerate form is honest.
\end{lemma}

\begin{proof}
	Assume first that $\mathcal H_F(p) = 0$. By  the Euler formula for homogeneous polynomials we have that
	$$
	(n-k)\partial_{i_1 \ldots i_k}F(x_0,\ldots,x_b)= \sum_{\ell=0}^b x_\ell \partial_{\ell, i_1 \ldots i_k}F(x_0, \ldots, x_b)
	$$
	for any $k=0,\ldots, n-1$ and $i_j \in \{0,\ldots, b\}$.
    Proceeding by induction and using the fact that $\mathcal H_F(p) = 0$, we get that $\partial_{i_1 \ldots i_k}F(p)=0$ for any $k=0,\ldots, n-1$ and $i_j \in \{0,\ldots, b\}$. Thus, $x_0$ does not appear in $F(x_0, \ldots, x_n)$.
	The converse is a simple computation.

\medskip

Assume now that $F(x_0, \ldots, x_n)$ is a non-degenerate form. If by contradiction there exists $p\in \mathbb P^b$ such that $\mathcal H_F(p) = 0$ then, up to a  change of coordinates, we may assume $p=[1,0,\ldots,0]$. This means that $F$ does not depends on $x_0$ and so the face of $\mathcal H_F(p)=0$  corresponding to $\partial_0$ is trivial. By \cite[Corollary XIV.1.5(d)]{GKZ}, it follows that $\det \mathcal H_F$ is identically zero, which is a contradiction.
\end{proof}

\begin{pmatrix}
0 & x_1   \\
x_1 & x_0   \\
\end{pmatrix}.
$$
By \cite[Proposition XIV.1.7]{GKZ}, it follows that $F_X$ is degenerate.
\end{example}

\section{Blow-ups}\label{s_blowups}

\begin{lemma}\label{lem:blowup}
Let $X$ be a smooth projective variety of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$.
Let $[E], \beta_1:=f^*\gamma_1, \ldots, \beta_b:=f^*\gamma_b$ be a basis of $H^2(Y,\Q)$ where $E$ is the exceptional divisor and $\gamma_1,\ldots,\gamma_b$ is a basis of $H^2(X,\Q)$.
Then, with respect to this basis, we may write
$$
F_Y(x_0,\ldots,x_n)=ax_0^n +\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \ldots, x_b)
$$
where $a=E^n$ and $R_i \in \mathbb Q[x_1,\ldots,x_b]$ is a form  of degree $i$, for $i=1,\dots,n-k$.

\end{lemma}
\begin{proof}
This follows easily from the projection formula.
\end{proof}

\begin{proposition}\label{prop:blowuprank}
Let $X$ be a smooth K\"ahler manifold of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \le 2$. Let $p=[E] \in H^2(Y,\C)$ be the class of the exceptional divisor.
\begin{enumerate}
\item \label{k=0} If $n\ge 2$ and $k=0$ then $\rk \mathcal H_{F_Y}(p)=1$.
\item \label{k=1}  If $n\ge 3$ and  $k=1$ then $\rk \mathcal H_{F_Y}(p) \ge 2$.
\item \label{k=2} If $n\ge 4$ and $k=2$ then, after a base change, we may write
$$
F_Y(x_0,\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\ldots,x_b) + x_0^{n-2}Q(x_1,\ldots,x_b) + F_X(x_1, \ldots, x_b)
$$
where $a \in \mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\rk \mathcal H_{F_Y}(p) \ge 2q$.

\end{enumerate}
\end{proposition}
\begin{proof}
We use the same notation as in Lemma \ref{lem:blowup}.

We first prove \eqref{k=0}. We have
$$
F_Y(x_0,\ldots,x_n)=ax_0^n  + F_X(x_1, \ldots, x_b)
$$
where $a=E^n \ne 0$. Hence $\rk \mathcal H_{F_Y}(p)=1$.
Thus, \eqref{k=0} follows.
\medskip

We now prove \eqref{k=1}. We have
$$
F_Y(x_0,\ldots,x_n)=ax_0^n+ x_0^{n-1}\cdot \left(\sum_{i=1}^{b}\lambda_ix_i \right )  + F_X(x_1, \ldots, x_b )
$$
where $a=E^n$ and $\lambda_i=Z \cdot \gamma_i$, for $i=1,\dots,b$. After taking a base change that fixes $(1,0,\ldots,0)$, we may assume $\lambda_i=0$ for any $i=2, \ldots ,b$, i.e.

$$
F_Y(x_0,\ldots,x_n)=ax_0^n+ \lambda_1 x_0^{n-1}x_1  + F_X(x_1, \ldots, x_b ),
$$
where $\lambda_1 \ne 0$ because we are blowing up a curve $Z$ in a K\"ahler manifold and so there exists at least one class in $H^2(X,\mathbb Z)$ with non-zero intersection with $Z$. Thus, it is easy to check that the subtensor given by
$$(\partial_0^{n-3}\partial_i\partial_j)_{i,j=0,\dots,b}$$
has rank two,  which implies that $\rk \mathcal H_{F_Y}(p) \ge 2$.

\medskip

We finally prove \eqref{k=2}. Let $j\colon Z \hookrightarrow X$ be the inclusion.
Let  $V \subset H^2(Z, \C)$ be the subspace generated by $j^*\gamma_1, \ldots, j^*\gamma_b$ and let $q$ be the rank of the quadratic form $Q$ obtained restrincting to $V$ the quadratic form of $H^2(Z,\mathbb C)$.
 Note that $q \ge 1$ because a K\"ahler class of $X$ restricts to a K\"ahler class on $Z$.
After a base change that diagonalises $Q$, we can write
$$
F=F_Y(x_0,\ldots,x_b)=\frac{ax_0^n}{n!} + \frac{x_0^{n-1}}{(n-1)!}(c_1x_1+\ldots +c_bx_b) + \frac{x_0^{n-2}}{2(n-2)!}(x_1^2+\ldots+ x_q^2) + F_X(x_1, \ldots, x_b)
$$
for some    $c_1,\dots,c_b \in \C$.

Consider the $(b+1)\times (b+1) \times (b+1)$ subtensor $A$ of $\mathcal H_{F}(p)$ given by the slices $A_0, \ldots, A_b$ where
 $$
A_h= ( \partial^{n-4}_0 \partial_h \partial_i \partial_j F(p))_{i,j=0,\dots,b}
 $$
 for $h=0,\dots,b$.

It is enough to prove that $\rk A \ge 2q$, as it immediately implies that $\rk \mathcal H_{F}(p) \ge 2q$.
We have
$$
A_0 =
\begin{pmatrix}
1 & c_1 & \ldots & c_q & c_{q+1} & \ldots & c_b \\
c_1 & 1  & 0 & \ldots & \ldots & \dots & 0 \\
\vdots & 0 & \ddots  & 0 & \cdots & \cdots & \vdots \\
c_q & 0 & \ldots & 1 & 0 & \ldots & 0  \\
c_{q+1} &\vdots & \vdots  &\vdots & 0 & \ldots & 0 \\
\vdots&\vdots & \vdots  &\vdots & \vdots & \ddots & \vdots \\
c_b & \cdots & \cdots & \cdots & 0 & \cdots & 0 \\
\end{pmatrix}

\label{s_volume}

The goal of this section is to present some open problems on the relationship between birational invariants and topological invariants of a smooth complex projective variety.

\sub

Let $X$ be a compact K\"ahler manifold of dimension $n$. One of the fundamental objects associated with $X$ is the natural form $F_X$ on $H^2(X,\mathbb{Z})$ of degree $n$, induced by the cup product structure. This form serves as a topological invariant of $X$ and plays a crucial role in understanding the intersection theory of higher-dimensional algebraic varieties.

For $n=2$, the quadratic form $F_X$ is central to the theory of algebraic surfaces, yielding results in classification theory and the study of their moduli spaces. When $n=3$, the study of the cubic form $F_X$ was initiated in \cite{OV95}, leading to numerous applications in algebraic geometry, differential geometry, and dynamical systems. Recent advancements have further deepened our understanding of $F_X$ and its role in various geometric problems \cite{KW, BCT16, CT18, ST19, ST20, Wilson21, Wilson21b}.

Despite these advances, relatively little is known about the structure and applications of $F_X$ in dimensions $n \geq 4$. The goal of this paper is to initiate a study of $F_X$ in higher dimensions, exploring its fundamental properties and potential applications to birational geometry and topology. In particular, we examine how the rank of the hypermatrix $\mathcal H_{F_X}$ of the $(n-1)$-th derivatives of $F_X$ influence the behavior of divisorial contractions (see Section \ref{s:intersection} for the definition of $\mathcal H_{F_X}$ and some of its basic properties).

\medskip

Our main result establishes the existence of a finite set of cohomology classes that control the exceptional divisors of divisorial contractions:

\label{s:intersection}

Let $M$ be a closed (i.e.\ compact and without boundary) oriented topological manifold of dimension $d=2n$.
Consider the symmetric
$n$-multilinear map defined by the cup prod

Our main result establishes the existence of a finite set of cohomology classes that control the exceptional divisors of divisorial contractions:

This result provides a topological bound on the possible exceptional divisors arising from some of the steps of the Minimal Model Program, establishing a new link between topology and birational geometry. It is a consequence of Theorem \ref{thm:W_F}, where we prove the finiteness of rank one points of $\mathcal H_F$ where $F$ is a non-degenerate $n$-form, a result of independent interest.

The structure of the paper is as follows. In Section \ref{s_preliminaries}, we introduce the necessary background on tensors and forms. Sections \ref{s_low} and \ref{s:intersection} are devoted to the study of the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$ with $n\ge 3$. We show, in particular, that, up to a non-zero scalar, there are at most finitely many classes $e\in H^2(M,\mathbb C)$ such that
$e^n\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.
(cf. Theorem \ref{thm:W_F}). Section \ref{s_blowups} establishes our main theorem by examining the rank of the associated form in the presence of a divisorial contraction. In Section \ref{s_volume}, we present some related open problems.

\subsection{Tensors} Let $a_1,\dots,a_n$ be positive integers. A \emph{tensor} of type $a_1\times \ldots \times a_n$ is a multilinear map
$$T\colon \mathbb C^{a_1}\times \ldots \times \mathbb C^{a_n}\to \mathbb C.$$
For any positive integer $a$, denote $[a]:=\{ 1,2,\ldots, a \}$. Then a tensor $T$ of type $a_1\times \ldots \times a_n$ is determined uniquely by a function
$$\tilde T\colon [a_1]\times \cdots \times [a_n] \to \mathbb{C}.$$
We will refer to $\tilde T$ as the \emph{hypermatrix associated to $T$}.
Given positive integers
$a_j^{'} \le a_j$ for $1 \le j \le n,$ and strictly increasing functions $f_j\colon [a_j^{'}] \to [a_j]$,
we define a \emph{sub-tensor} $T'$ of type $a'_1\times\ldots\times a'_n$ so that if $\tilde T'$ is the hypermatrix associated to $T'$ then
$$\tilde T'=\tilde T(f_1,\dots,f_n)\colon [a'_1]\times \cdots \times [a'_n] \to \mathbb{C}.$$
Note that if $v_i\in \mathbb C^{a_i}$ for $i=1,\dots,n$, then there exists a natural tensor of type $a_1\times \ldots \times a_n$ defined by
$v_1\otimes \ldots \otimes v_n$.

\begin{lemma}\label{lem:blowup}
Let $X$ be a smooth projective variety of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$.
Let $[E], \beta_1:=f^*\gamma_1, \ldots, \beta_b:=f^*\gamma_b$ be a basis of $H^2(Y,\Q)$ where $E$ is the exceptional divisor and $\gamma_1,\ldots,\gamma_b$ is a basis of $H^2(X,\Q)$.
Then, with respect to this basis, we may write
$$
F_Y(x_0,\ldots,x_n)=ax_0^n +\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \ldots, x_b)
$$
where $a=E^n$ and $R_i \in \mathbb Q[x_1,\ldots,x_b]$ is a form of degree $i$, for $i=1,\dots,n-k$.

\begin{proposition}\label{prop:blowuprank}
Let $X$ be a smooth K\"ahler manifold of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \le 2$. Let $p=[E] \in H^2(Y,\C)$ be the class of the exceptional divisor.
\begin{enumerate}
\item \label{k=0} If $n\ge 2$ and $k=0$ then $\rk \mathcal H_{F_Y}(p)=1$.
\item \label{k=1} If $n\ge 3$ and $k=1$ then $\rk \mathcal H_{F_Y}(p) \ge 2$.
\item \label{k=2} If $n\ge 4$ and $k=2$ then, after a base change, we may write
$$
F_Y(x_0,\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\ldots,x_b) + x_0^{n-2}Q(x_1,\ldots,x_b) + F_X(x_1, \ldots, x_b)
$$
where $a \in \mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\rk \mathcal H_{F_Y}(p) \ge 2q$.

\begin{example}
Let $X$ be a complex projective manifold of dimension $n$ and $b_2(X)=1$. Let $f\colon Y \to X$ be the blow-up along a smooth curve $Z$ of $X$. Let $p=[E] \in H^2(Y,\C)$ be the class of the exceptional divisor. Then we can write
$$
F_Y(x_0,\ldots,x_n)=\frac{a}{n!}x_0^n+ \frac{x_0^{n-1}x_1}{(n-1)!} + x_1^n,
$$
for some $a \in \mathbb Q$. If $n=3$, then
$$
\mathcal H_{F_Y}(p)=
\begin{pmatrix}
a & 1 \\
1 & 0 \\
\end{pmatrix}
$$
has rank 2. If $n=4$, then
$$
\mathcal H_{F_Y}(p)=
\left[ \begin{pmatrix}
a & 1 \\
1 & 0 \\
\end{pmatrix},
\begin{pmatrix}
1 & 0 \\
0 & 0 \\
\end{pmatrix}
\right]
$$
has rank at least 3 by Lemma \ref{lem:trick}. On the other hand, 3 is the maximal rank for a $2\times 2 \times 2$ tensor (see, for instance, \cite[Section 3]{Kruskal}). Thus, $\mathcal H_{F_Y}(p)$ has rank $3$.
\end{example}

\begin{question}\label{q:volume}
Let $M$ be a closed topological manifold of dimension $2n$. Is there a constant $C$, depending only on $M$, such that for any smooth complex projective variety $X$ whose underlying topological space is $M$, we have
\[ \vol(X) \le C? \]
\end{question}

The proof of Theorem~\ref{thm_main} is obtained by reducing it to a finiteness statement about certain cohomology classes detected by the intersection form. Concretely, the paper studies “the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$,” and shows that “up to a non-zero scalar, there are at most finitely many classes $e\in H^2(M,\mathbb C)$ such that $e^n\neq 0$ and the rank of the Hessian of $F$ at $e$ is one” (cf. Theorem~\ref{thm:W_F}), which itself comes from “the finiteness of rank one points of $\mathcal H_F$ where $F$ is a non-degenerate $n$-form.” Then Section~\ref{s_blowups} “establishes our main theorem by examining the rank of the associated form in the presence of a divisorial contraction,” relating the exceptional divisor class $[E]$ of a contraction $f\colon X\to Y$ to one of those finitely many rank-one Hessian classes, yielding the finite list $e_1,\ldots,e_q$.

The proof of Theorem~\ref{thm_main} is obtained by reducing it to a finiteness statement about certain cohomology classes detected by the intersection form. Concretely, the paper studies “the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$,” and shows that “up to a non-zero scalar, there are at most finitely many classes $e\in H^2(M,\mathbb C)$ such that $e^n\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.” We first record the precise finiteness statement:

\begin{theorem}\label{thm:W_F}
Let $F \in \C[x_0, x_1, \ldots,x_b]$ be a form of degree $n \ge 3$.
\begin{enumerate}
\item If $F$ is honest, then $W_F \cap \{F \ne 0\}$ is a finite set, and more precisely,
$$
|W_F  \cap \{F \ne 0\} | \le b+1.
$$

\item If $F$ is non-degenerate, then $W_F$ is a finite set.
\end{enumerate}
\end{theorem}

This itself comes from “the finiteness of rank one points of $\mathcal H_F$ where $F$ is a non-degenerate $n$-form.” Next we study blow-ups and use them to connect exceptional divisor classes to these finitely many rank-one Hessian classes. In particular, we use the following structural description of the intersection form under blow-up:

\begin{lemma}\label{lem:blowup}
Let $X$ be a smooth projective variety of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$.
Let $[E], \beta_1:=f^*\gamma_1, \ldots, \beta_b:=f^*\gamma_b$ be a basis of $H^2(Y,\Q)$ where $E$ is the exceptional divisor and $\gamma_1,\ldots,\gamma_b$ is a basis of $H^2(X,\Q)$.
Then, with respect to this basis, we may write
$$
F_Y(x_0,\ldots,x_n)=ax_0^n +\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \ldots, x_b)
$$
where $a=E^n$ and $R_i \in \mathbb Q[x_1,\ldots,x_b]$ is a form  of degree $i$, for $i=1,\dots,n-k$.

\end{lemma}

We then control the rank of the Hessian at the exceptional divisor class $p=[E]$ in low-dimensional centers of blow-up via the following result:

\begin{proposition}\label{prop:blowuprank}
Let $X$ be a smooth K\"ahler manifold of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \le 2$. Let $p=[E] \in H^2(Y,\C)$ be the class of the exceptional divisor.
\begin{enumerate}
\item \label{k=0} If $n\ge 2$ and $k=0$ then $\rk \mathcal H_{F_Y}(p)=1$.
\item \label{k=1}  If $n\ge 3$ and  $k=1$ then $\rk \mathcal H_{F_Y}(p) \ge 2$.
\item \label{k=2} If $n\ge 4$ and $k=2$ then, after a base change, we may write
$$
F_Y(x_0,\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\ldots,x_b) + x_0^{n-2}Q(x_1,\ldots,x_b) + F_X(x_1, \ldots, x_b)
$$
where $a \in \mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\rk \mathcal H_{F_Y}(p) \ge 2q$.

\end{enumerate}
\end{proposition}

In establishing the main theorem, we apply these blow-up computations to examine “the rank of the associated form in the presence of a divisorial contraction,” relating the exceptional divisor class $[E]$ of a contraction $f\colon X\to Y$ to one of the finitely many rank-one Hessian classes provided by Theorem~\ref{thm:W_F}, yielding the finite list $e_1,\ldots,e_q$.

Question: Let $M$ be a closed (compact and without boundary) topological manifold of real dimension $2n$, and set $b=\dim_{\mathbb C} H^2(M,\mathbb C)$. For any smooth complex projective variety $X$ of complex dimension $n$ whose underlying topological space is $M$, identify $H^2(X,\mathbb C)$ with $H^2(M,\mathbb C)$. If $f\colon X\to Y$ is a divisorial contraction to a point, let $E$ denote its exceptional divisor and let $[E]\in H^2(M,\mathbb C)$ be its cohomology class under this identification. Which existence statement holds?

There exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ with $q\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\colon X\to Y$ to a point with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]=e_i$.

Choices:
B. There exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ with $q\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\colon X\to Y$ with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]=e_i$.
C. There exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ for some finite $q$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\colon X\to Y$ to a point with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]=e_i$.
D. For every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, there exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ with $q\le b+1$ such that for every divisorial contraction $f\colon X\to Y$ to a point with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]=e_i$.
E. There exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ with $q\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\colon X\to Y$ to a point with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]$ is proportional to $e_i$ by a nonzero scalar.