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One of my personal goals for 2015 was to attend a sewing retreat. To that end, I've signed up to attend Sewtopia in Chicago next month (April). There's more info on Sewtopia below.
Now that we are less than a month out, I'm starting to work on a few projects for the retreat. First on my list was to create something (anything - there were no restrictions!) for a swap.
For this project, everything that I used came from my scrap bins, except for the black and white stripe fabric. I had a lot of fun digging through my pink, red and orange scrap bins. Since the block is so bright and busy, and the background is full of words (Allison Glass Sun Print Text), I opted to keep the quilting simple and just stitched in the ditch around the red wings, and then did 1 "echo" line around each wing in white (hard to see, but it's there in the above photo - and is clearer in the photo below).
For fun, I added one of my labels from my fabric gift bags and called it done!
Two other projects for Sewtopia, involved picking fabrics for the two classes that I will be taking with Latifah Saafir from Latifah Saafir Studios and Rita Hodge from Red Pepper Quilts.
Latifah is teaching us to make her molehill quilt (curved piecing - upper left). Above (right) are the fabrics that I've picked out.
Rita is going to work with us on making a Kansas Dugout mini-quilt. I had fun using Electronic Quilt (design program below) to try out different color and fabric schemes for this one.


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"Pray if you want to:" a reevaluation of religion in the fiction of Ernest J. Gaines
Evelyn E. Kelly (Creator)
SallyAnn Ferguson
Abstract: This study examines religiosity in Ernest J. Gaines's fiction. Though some critics acknowledge the existence of religion in Gaines's work, most of them gloss over the topic. Generally, they conclude that the author mistrusts religion, religion fails the characters, and his preacher personas are perfunctory. I argue that such criticism fails to address fully the role of religion in Gaines's writing. Because of the ubiquitousness of religious circumstance, I argue that Gaines expresses neither rejection of nor ambivalence towards religion. Instead, I offer evidence that his stance is paradoxical. Characters who are atheists or claim to have lost faith are the characters who most epitomize Christian traditions and characteristics. Many use preacherly techniques such as speech and mannerisms, and others exemplify biblical characters through parallel lifestyles. Though several characters claim to reject religion, Christian principles impact and guide their behavior and thought. Additionally, Gaines inserts preacher personas whose words and actions seem to support the notion that the men are perfunctory; however, their roles as motivators and activists suggest otherwise. Gaines also acknowledges African American folk tradition with his insertion of folkloric preachers and conjurers. Though Gaines's primary themes are manhood issues, I argue that religion plays a role in creating and maintaining a sense of manhood. To analyze Gaines's treatment of religion, I discuss his use of four elements of narratology: the imagery of falling, rhetorical manipulation, the dismantling of stereotype, and the incorporation of biblical allusion and typological symbolism and archetypes. My primary argument is that Ernest Gaines promotes orthodox Christianity by presenting it in an acceptable modern form. Utilizing a range of characters who exhibit varied attitudes toward religion, from unquestioning acceptance to outright rejection, Gaines "seemingly" advocates negative attitudes toward Christianity, but this exploitation of the negative repeatedly serves as a segue to reveal a positive update. Located in the background of each story is a positive historical or cultural factor related to religion, proving that religion is an escapable, fundamental aspect of African American culture.
African American, Christianity, conjure, manhood, preacher, tradition
Gaines, Ernest J., $d 1933- $x Criticism and interpretation.
Christianity in literature.
Masculinity in literature.
African Americans in literature.

redpajama_set_name: RedPajamaCommonCrawl


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text:
Ron Marinucci October Column
by Ron Marinucci, Oct. 1, 2016
Provided by RunMichigan.com
In mid-June my friend Carrie Farnum and I were running out on a back road, just slogging some miles. Cresting a slight hill, in the distance we could see another runner approaching. "He's running pretty fast," Carrie noted.
As we closed the gap, I recognized the other runner. It was Doug Goodhue and he was "running pretty fast." Meeting, we stopped and exchanged greetings and an introduction ("Carrie-Doug, Doug-Carrie") for a few moments. Goodhue explained that he was tuning up for the Volkslaufe 20K to be held in Frankenmuth in a few weeks, getting in a hard 10-miler, "a tempo run," he said. He had his eyes set on the 70-74 American age-group record at that distance. Considering the heat and humidity of that morning, along with those hilly back roads, and at his pace, this was a good, tough tune-up for the Volkslaufe.
Getting in our so-longs and running in opposite directions, Carrie repeated, "He was fast!" She was right and I added, "He still is."
The "tempo run" of mid-June must have worked because a couple weeks later, Goodhue did indeed set the US age-group record at the Volkslaufe. His 20K time of 1:29:28 was almost three full minutes faster than the previous best.
Longtime Michigan runners know that Goodhue is no stranger to running records, championships, and other honors. He has held or still holds dozens of age-group and single-age records. That's records, not merely age-group wins. His first win came in 1985 and, as of July 2016, he noted, "My next age-group win will be my 430th."
He's also carted away 45 USA Track and Field individual national championships. Toss in another twelve USATF team national championships running
with the Ann Arbor Track Club. Since turning 60, he has been ranked number one nationally in his age-group nine times, including for eight consecutive years.
The Road Runners Club of America named him the 2010 Male Master Runner-of-the-Year and last year he was inducted into the USATF Masters Hall of Fame.
Looking back, he ran cross country and track in Detroit at Redford High School. When he graduated in 1960, the running stopped, at least for a while—a long while. Attending Lawrence Tech and getting established with family and career put running on hold.
Twenty-two years later, in 1982, "I returned to running," he said. The next year he entered his first race, the 1983 Detroit Free Press Marathon. He finished, he admitted, in "four hours plus." But that certainly was no indication of what was to come.
In 1985, at the Plymouth YMCA 5K, he scored that first age-group win. His 18:16 topped the 40-49 age-bracket. And the wins just kept coming.
Goodhue has been and remains a versatile runner, enjoying and succeeding in a variety of distances and courses. He's won and set records in cross country as well as on the track and on the roads. Although he hasn't run a marathon in many years, he also has held records at a variety of distances. For example, in addition to topping the 20K record books in both the 65-69 and 70-74 age-groups, he's also broken records at 3000 meters indoors on the track, 5K on the roads, and 10K cross country events. Since turning 70, he's run a mile under six minutes and a 5K under twenty minutes, while finishing a 10K in just over 40 minutes.
The retired father of four confided, "I still wake up every morning thinking about my next competition, my next challenge!"
He regularly puts in about 50 miles of training each week. It includes "speed work on the track, hill repeats, and tempo runs," like the day Carrie and I crossed running paths with him. "All [these workouts] are geared toward my running schedule. I train to race!"
As noted, Goodhue hasn't run a marathon in many years and even now has cut back on his racing, if only a bit. "Because of a number of nagging lower leg injuries over the last few years, I have had to reduce the number of races I run." "Reduce" does not mean stop, nor does it mean diminished quality.
He's not just about running, but includes the running community, too. He's been a member of the Ann Arbor Track Club for more than 30 years. In his fifth year as a certified race director, he helps to direct several high-profile area races for the club, including Dexter-Ann Arbor. He's also a USATF and RRCA certified coach. He added, "I just completed my 10th year of coaching the Running Fit 501 Training Team," along with co-coach Suzi Stock, Lee Mamola, and Goodhue's wife, Cindy.
Looking ahead, in July he said, "My plans for the rest of 2016 and the last year in my present age-group [70-74], I plan to run the Crim 10 Miler," in which he finished with an age-group third [1:26:43], "and the rest of the Masters Grand Prix Series."
Competitors and records beware!


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Quadrotor automatically recovers from failure or aggressive launch, without GPS
by Matthias Fässler, Flavio Fontana, Elias Müggler, Christian Forster and Davide Scaramuzza
Photo credit: Robotics & Perception Group, University of Zurich.
When a drone flies close to a building, it can temporarily lose its GPS signal and position information, possibly leading to a crash. To ensure safety, a fall-back system is needed to help the quadrotor regain stable flight as soon as possible. We developed a new technology that allows a quadrotor to automatically recover and stabilize from any initial condition without relying on external infrastructure like GPS. The technology allows the quadrotor system to be used safely both indoors and out, to recover stable flight after a GPS loss or system failure. And because the recovery is so quick, it even works to recover flight after an aggressive throw, allowing you to launch a quadrotor simply by tossing it in the air like a baseball.
Our quadrotor is equipped with a single camera, an inertial measurement unit, and a distance sensor (Teraranger One). The stabilization system of the quadrotor emulates the visual system and the sense of balance within humans. As soon as a toss or a failure situation is detected, our computer-vision software analyses the images for distinctive landmarks in the environment, and uses these to restore balance.
All the image processing and control runs on a smartphone processor on board the drone. The onboard sensing and computation renders the drone safe and able to fly unaided. This allows the drone to fulfil its mission without any communication or interaction with the operator.
The recovery procedure consists of multiple stages. First, the quadrotor stabilizes its attitude and altitude, and then it re-initializes its visual state-estimation pipeline before stabilizing fully autonomously. To experimentally demonstrate the performance of our system, in the video we aggressively throw the quadrotor in the air by hand and have it recover and stabilize all by itself. We chose this example as it simulates conditions similar to failure recovery during aggressive flight. Our system was able to recover successfully in several hundred throws in both indoor and outdoor environments.
More info: Robotics and Perception Group, University of Zurich.
M. Faessler, F. Fontana, C. Forster, D. Scaramuzza. Automatic Re-Initialization and Failure Recovery for Aggressive Flight with a Monocular Vision-Based Quadrotor. IEEE International Conference on Robotics and Automation (ICRA), Seattle, 2015.
M. Faessler, F. Fontana, C. Forster, E. Mueggler, M. Pizzoli, D. Scaramuzza. Autonomous, Vision-based Flight and Live Dense 3D Mapping with a Quadrotor Micro Aerial Vehicle. Journal of Field Robotics, 2015.
Quadrocopter failsafe algorithm: Recovery after propeller loss
Video: Throwing and catching an inverted pendulum – with quadrocopters
Developing trust in autonomous robots: Seminar with Michael Wagner
From sideshow to main event, drones are making their mark in entertainment
tags: c-Aerial, cx-Research-Innovation, flight recovery, quadrocopter, quadrotor, Robotics and Perception Group, Switzerland, University of Zurich, video
Matthias Fässler is a PhD student at Prof. Scaramuzza's Robotics and Perception Group of the University of Zurich.
Flavio Fontana is a PhD candidate at the Robotics and Perception Group.
Elias Müggler is a PhD student at Prof. Scaramuzza's Robotics and Perception Group of the University of Zurich.
Christian Forster is a PhD student at the Robotics and Perception Group.
Davide Scaramuzza is Assistant Professor of Robotics at the University of Zurich.


text:
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redpajama_set_name: RedPajamaStackExchange


tl;dr: CHI is growing so now CHI has a steering committee; 2017-2019 cities and chairs are in place; we are seeking chairs and venues for CHI 2020 as we return to Asia in 2021.
The past 33 years have seen many successes, developments and challenges for the CHI conference series - most of all due to its growing size and need for streamlined logistics. The first ACM SIGCHI sponsored CHI conference (Human Factors in Computing Systems) was held in 1983 with around 1,000 delegates where 59 papers were presented. Fast forward to today, where CHI 2016 had over 3500 delegates and at times 19 simultaneous tracks. Further, the conference has become more global, and more diverse in terms of topics, methods, and disciplines. As stewards of our community's flagship conference, we must ensure that the conference changes as necessary to meet these new opportunities and new challenges.
For CHI, the SIGCHI executive committee votes on who becomes the general chairs, technical program chairs, and papers chairs. However, unlike all the other conferences SIGCHI sponsors, CHI has not had a steering committee. Instead the organisation of each year's conference relied heavily on the Vice-President for Conferences, working closely with the General Chairs, Technical Program Chairs, organising committee and the ACM. This approach has enabled the successful CHI conferences we have all experienced, and has allowed SIGCHI to build up a wealth of knowledge and experience with running CHI. However, it also placed a large burden on a single volunteer to maintain support for CHI. Further it did not give the communal ownership and buy-in that a steering committee can. Therefore, to better share the volunteer load, to engage the community more deeply, and to ensure continuity of SIGCHI management of the CHI Conference, the SIGCHI Executive Committee (EC) is pleased to announce the formation of a CHI Conference Steering Committee.
The chair of the CHI steering committee will join the SIGCHI EC as Adjunct Chair for CHI. The responsibilities of the CHi steering committee were previously distributed across members of the EC. As a result, we will see changes within the EC itself and beyond. The operational aspects of our conferences, which the Vice President for Conferences formerly undertook for CHI and SIGCHI, will now become the responsibility of the Vice President for Operations. The strategic direction and operational oversight for CHI, and recommendations on such to the EC, which had been the responsibility of the Vice President for Conferences, now will rest with the Adjunct Chair for CHI. Finally, the SIGCHI Vice President for Conferences will have oversight for all SIGCHI conferences, including CHI, and thus will subsume all the responsibilities of the former SIGCHI Adjunct Chair for Specialised Conferences. Thus, the Adjunct Chair for Specialized Conferences role will end.
We also need some structural changes on various committees and SIGCHI boards that assist the vice presidents in their roles. Our intention is to clarify our bylaws and to form ad-hoc committees (rather than standing committees) to support our activities. Changes to our bylaws will first need to be approved by our membership prior to SIGCHI changing our way of working. These changes will allow the various groups to function effectively and at a scale which is suitable for making decisions to support our conferences, members, and community.
To achieve this, we have brought together a group of future CHI chairs, past CHI chairs, past technical program chairs, and past papers chairs to provide strategy, oversight and management for CHI, along with ACM staff who will act as liaison to the committee. The committee is now being formed and will start to meet regularly from December 2016. Over the coming year, members of the committee will observe and study how CHI is managed and the day to day challenges the organising committee face. The former Vice President for Conferences, Scooter Morris, is now a member of the CHI steering committee and will continue to be the liaison to CHI 2017 to maintain a sense of continuity.
Liaison teams have been identified for CHI 2018 and 2019 and as plans for CHI 2020 and 2021 come online, new volunteers from the committee will be identified to help advise each new organising committee. Our conferences are organised largely by volunteers but even they need help and advice when they run into challenges and problems. Our hope is that this new steering committee will afford a range of support all new chairs need as they plan for a successful CHI.
Between now and the CHI conference in 2019, the CHI steering committee will operate in a setup phase. This will allow the committee to determine how best to support each year's conference. The committee will provide advice to the SIGCHI EC during this period of time. After 2019, we expect the CHI steering committee to move into a more steady state of operation. The hope is that shared knowledge and experience can provide a bedrock of support as each future CHI starts its development and planning. We also aim to ensure the CHI steering committee has the necessary support to undertake day to day actions and develop strategy for future CHI conferences.
Beyond support for each year's CHI conference, there are many long-term and strategic questions this steering committee must face. The first and most obvious is the continued growth in paper submissions to CHI, which is not commensurate with growth in the qualified reviewer pool or the number of delegates who attend CHI. Aspects of the program which may have been intended as single-year innovations have become "baked" into the conference, and there is a perception that this limits the flexibility of the CHI steering committee and future CHI chairs to innovate. This steering committee will consider what is core to CHI and propose various models which can address concerns from the community, delegates and organisers. The steering committee has many other challenges to consider, including how we foster intellectual progress in the field of HCI, registration costs, submission process and deadlines, and future locations given the size and nature of CHI along with costs.
Siân Lindley (Microsoft Research Cambridge, UK) to replace Vicki Hanson as a Vice President at Large.
Philippe Palanque (Université Paul Sabatier, France) as Adjunct Chair for CHI.
chaired by Regan Mandryk (University of Saskatchewan, Canada) and Mark Hancock (University of Waterloo, Canada).
the 2018 Papers Chairs are mc schraefel (University of Southampton, UK), Ed Cutrell (Microsoft Research, India) and Anind Dey (CMU, USA).
chaired by Geraldine Fitzpatrick (TU Wien, Austria) and Stephen Brewster (University of Glasgow. UK).
How are the venues for CHI conferences selected?
One question SIGCHI members might have is: how are the venues for CHI conferences selected? CHI has been held in the USA 22 times, and outside the USA 10 times. As the new SIGCHI Steering Committee is formed, the question becomes, where do we cite future CHI conferences to best suit our global membership? Planning for the future is of foremost concern, to ensure timely venue identification. As such, 2020 will see the conference location 'heading towards' Asia while 2021 we plan to return to Asia.
Broadly speaking the proposed process for strategically planning for conference locations includes about ten stages spread over a few years. The start of this process is outlined in Figure 1. First, the SIGCHI executive committee develops strategic location requirements which are provided to the CHI Steering Committee. This committee takes this guidance, along with previous CHI data, CHI development forecasting, and community needs to develop a list of potential cities. The forecasts and prior knowledge of CHI feed into our meeting requirements which detail space and facilities needs for CHI. These details along with details of potential cities are moulded into a request for proposals which is sent directly from the ACM on behalf of SIGCHI to a small set of convention centres in the target location for a particular year of CHI.
This process is sometimes a surprise to SIGCHI members who might be familiar with specialised conferences which first select conference chairs or who take complete bids to host a conference. When CHI had a small number of delegates, and a wide range of venues could host the conference, this was possible. Today, the scale of CHI requires us to take a different approach as we try to align community needs with potential cities or convention centre availability along with our space requirements. Once detailed bids are returned from the potential cities, an assessment is undertaken, followed by visits to several potential sites. This leads to Figure 2, where an understanding of the pros and cons of each proposal are considered. These factors include dates, cost, space, accessibility, etc. The CHI steering committee considers this broad range of factors in making a site recommendation to the SIGCHI Executive Committee.
At this stage you might think the planning work for a particular CHI is done? In reality, that is not the case. Next, ACM needs to complete contract negotiation with the site the SIGCHI EC selects. Given the complexity of our requirements, this can be a time-consuming process. In parallel, conference chairs must be identified. Sometimes the chairs for a future CHI are identified around the time potential cities are under consideration but in reality these activities are decoupled to ensure the final site selection process can focus on a fair assessment of the site.
Today, there are hundreds of volunteers involved in the organisation of any one CHI conference, or thousands if you count all the authors and reviewers who give of their time. CHI is also many things to many people. Aspects of the program which are held dear by one part of the community are seen as a luxury to others. What delegates from different types of industries, different academic traditions or different approaches to working with clients want vary considerably. Each year, the CHI chairs need to balance the competing interests, the available space, the budget, etc. as they plan and organize what they hope will be the best CHI ever.
In short, the steering committee for CHI needs to marry the myriad of interests in CHI to ensure it remains the flagship event for the global HCI community.
We hope you join us in offering best wishes and support to the new CHI Steering Committee, and CHI 2017, 2018 and 2019 organizing committees.


Dear friends of Hog Island: I am very pleased to share the news that we have enjoyed three very successful weeks at Hog Island this past June. The program this year operated under Project Puffin (which is part of National Audubon Society's Science Division). There were three sessions — all focusing on birds.


text:
Q: how to make a button of free shape? I want to create map app. For example, the are different areas in Japan. All of them are different shape, not square. You push on it and it gonna change the color. It is possible to create the button of unusual form (the form of region on the map) in Xcode?

A: Drawing
In order to draw the custom shape on the map, use your custom subclass of MKPolygon where you'll define your selected state color, and selected flag.
internal final class MyPolygon: MKPolygon {
   internal var selected = false
   internal var selectedColor = UIColor.red.withAlphaComponent(0.6)

   ...
}

In order to draw a custom color for your custom MKPolygon, create your subclass of MKPolygonRenderer  and use custom drawing to fill the polygon with the selectedColor when it has selected == true.
When those two implemented, add your overlay onto MKMapView
mapView.addOverlay(polygon)

Assign the delegate to your mapView, and return your renderer for the polygon from the delegate's method:
func mapView(_ mapView: MKMapView, rendererFor overlay: MKOverlay) -> MKOverlayRenderer {
    if let polygon = overlay as? MyPolygon {
        return MyPolygonRenderer(polygon: polygon)
    }
    return MKOverlayRenderer(overlay: overlay)
}

Catching user input
Add UITapGestureRecognizer onto your mapView:
func viewDidLoad() {
    super.viewDidLoad()
    let tapRecognizer = UITapGestureRecognizer(target: self, action: #selector(mapViewDidTap(_:)))
    mapView.addGestureRecognizer(tapRecognizer)
    ...
}

func mapViewDidTap(_ tapRecognizer: UITapGestureRecognizer) {
    if tapRecognizer.state == .recognized {
        // Get map coordinate from touch point
        let touchCoord = tapRecognizer.location(in: mapView)
        let coord = mapView.convert(touchCoord, toCoordinateFrom: mapView)
        let coordRect = MKMapRect(origin: coord, size: MKMapSize())
        // for every overlay ...
        for overlay in mapView.overlays {
            // .. if MyPolygon ...
            if let polygon = overlay as? MyPolygon {
                let selected = polygon.intersects(coordRect)
                if polygon.selected != selected {
                    polygon.selected = selected
                    mapView.renderer(for: polygon)?.setNeedsDisplay()
                }
            }
        }
    }
}


Merry Christmas from Prosthetic & Orthotic Associates!
Wishing our POA family and friends a Merry Christmas and Happy New Year! We count our blessings every day, and you are on top of that list.Thank you for entrusting your prosthetic care to us. The following video features just a few of the many, but ALL of you are in our hearts!


text:
\section{Introduction}
Let $k$ be a field and $X$  an algebraic variety over $k$. The group of birational transformations on $X$ is denoted by $\Bir_k(X)$ or just $\Bir(X)$.
Besides a rich history of 160 years of work on this group, there are many mysteries about this group even in the case where $X=\mathbb{P}^n$.
For the group theoretic structures, there are several good classical results for $\Bir(\mathbb{P}^2)$, since Noether and Castelnuovo. Nevertheless,  the non-simplicity and some strange behaviors of  normal subgroups were  just discovered in 2013 by Cantat and Lamy \cite{CL13}. $\Bir(\mathbb{P}^n)$ is more involved and less is known about the group structure of $\Bir(X)$.

In 2010 J.-P. Serre \cite{Se10} introduced a topology on $\Bir(X)$ through  morphisms from algebraic varieties.
In 2013, in their seminal work, Blanc and Furter prove that
\begin{quote}
For any $n \geq 2$, there is no structure of  an algebraic variety of infinite dimension on $\Bir(\mathbb{P}^n)$ such that families $A \to \Bir(\mathbb{P}^n)$ correspond to morphisms of algebraic varieties.
\end{quote}
In the same paper, Blanc and Furter prove that for any fixed integer $d$, the set of Birational transformation of  degree $d$ (polynomial degree $d$ without common factor), $\Bir(\mathbb{P}^n)_d$ has the structure of an algebraic variety, even though, $\Bir(\mathbb{P}^n)_{\leq d}$ cannot be viewed as an algebraic variety for $d, n \geq  2$.

The algebraic structure of $\Bir(\mathbb{P}^2)_d$, led to  the classification of the irreducible components of this variety by Bisi, Clabri, and Mella in 2015 \cite{BCM15}.

In 2017, in  the survey \cite{B17}, Blanc bolded the question:
\begin{center}
What kind of algebraic structure can we put on Bir(X)?
\end{center}
In this note, we show that for an irreducible  projective variety $X$ over any field  $k$ and any integer $d$, $\Bir_k(X)_d$ accepts the structure of a constructible algebraic set. To establish the result in full generality, that is, for any projective variety $X$ in any characteristic, we reconcile several openness properties in commutative rings. That includes openness of the {\it  ideals of principal class}, Lemma \ref{Elimination}, the openness of {\it  regular sequences}, Lemma \ref{LG2}, and openness of {\it  ideals with maximal analytic spread}, Lemma \ref{LNn+1}.  There are two new concepts in this note. One is the definition of the {\it clear polynomial degree} of a rational map, Definition \ref{Ddegree}, that is the first subtlety to pass from a factorial variety $X$ to an arbitrary variety. The second  one is the $d$'th {\it Elimination index} of $X$, $\epsilon_X(d)$, which measures how far from $\mathbb{P}^n$ the variety $X$ is. Corollary \ref{C1}, states that in the case where $\epsilon_X(d)=0$,  $\Bir(X)_d$ is  a quasi-projective variety. In particular,  $\Bir(\mathbb{P}^n)_d$ is a quasi-projective variety for all $d$.

\section{Main Result}
 We fix a field $k$, of any characteristic and not necessarily algebraically closed, and a projective variety $X\subseteq \mathbb{P}^n_k$ reduced and irreducible.   We assume $X$ is  non-degenerated i.e. $X$ is not contained in any hyperplane, and denote  the coordinate ring of $X$ by $R$.

 A rational map from $X$ to $X$ is given by $$h:(x_0:\cdots:x_n)\mapsto (h_0(x_0:\cdots:x_n):\cdots:h_n(x_0:\cdots:x_n)).$$
 Here $h_0,\cdots, h_n$ belong to $R$ and  all of  them are of the  same degree. One faces preliminarily the fact that the map can be defined by different sets of forms of fixed degree, where this degree may vary from set to set.  We present two examples:
 \begin{Example} \label{Ex1}  Let consider the rational maps
  \begin{align*}
\sigma_i:X_i&\dasharrow X_i\\
(x:y:z)&\mapsto (yz:xz:xy)
\end{align*}
in the following two cases
 \begin{itemize}
 \item{ $X_1=V(y^2-xz)$, a smooth curve in $\mathbb{P}^2$, and }
 \item{$X_2=V(y^3-x^2z)$ a cuspidal curve ($(S_2)$ not $(R_1)$), in $\mathbb{P}^2$.}
 \end{itemize}
 $\sigma_1$  is also represented by $(z:y:x)$; while $\sigma_2$ admits another representative of degree $2$ that is not obtained by multiplication, namely $\sigma_2$ is the same as the map \footnote{ We also notice that the map defined by the coordinates of $\tau$ does not belong to $\Bir(\mathbb{P}^2)$ while that of $\sigma_2$ belongs to $\Bir(\mathbb{P}^2)$, as well, any other representative of $\sigma_2$ is a combination of $\sigma_2$ and $\tau$.}

 \begin{align*}
\tau:X_2&\dasharrow X_2\\
(x:y:z)&\mapsto (xz:y^2:x^2).
\end{align*}
 \end{Example}

 One way to circumvent the subtlety of defining the degree of a birational map is to look at the grade of the ideal of the base of the map. Recall  that, for an ideal $I$ in the ring $R$, $\grade(I,R)$ is the maximum length of a proper regular sequence inside $I$. Consider the map $h$, above. Let $I=(h_0,\ldots,h_n)\subset R$.  If $\grade(I,R)\geq 2$, then the map $h$ is presented, uniquely, up-to multiplication into an element of $R$ (and not in $Q(R)\setminus R$), c.f.   \cite[Proposition 1.1]{Si04}.
 Accordingly, we set the following definition:

 \begin{Definition}\label{Ddegree} Let $h:X\dasharrow X$ be a rational map.  We say the map $h$ has {\it clear polynomial  degree} $d$, if $h$ admits a representative  $(h_0:\cdots:h_n)$ such that  the ideal in $R$ generated by $h_0,\ldots,h_n$ has grade at least $2$ and  $\deg(h_i)=d$.
  \end{Definition}
  One says  $X$ satisfies the Serre's $(S_2)$-condition if  for all prime ideal $\mathfrak{p}\subset R$,
$\depth(R_{\mathfrak{p}})\geq \min\{\codim(\mathfrak{p}),2\}.$ In the case where $X$ is $S_2$ or at least $S_2$ in codimension $2$, then   $\grade(I,R)\geq 2$ if and only if $\codim(I)\geq 2$.  In particular, if $R$ is factorial, then $\grade(I,R)\geq 2$ if and only if $h_0,\ldots,h_n$ have no common factor.

   \begin{Definition}\label{DBird} We denote by $\Bir(X)_d$ the set of birational maps from $X$ to $X$  of clear polynomial degree $d$.  $\Bir(X)_{\ast}:=\cup_{d=1}^{\infty}\Bir(X)_d$.   If, for example, $R$ is factorial, then $\Bir(X)_{\ast}=\Bir(X)$.
\end{Definition}
In Example \ref{Ex1}, $\sigma_1\in \Bir(X_1)_1\setminus \Bir(X_1)_2 $ while $\sigma_2\in \Bir(X_2)\setminus \Bir(X_2)_*$.

\begin{Remark}  Once we study the whole group $\Bir(X)$, it is  possible, and  indeed useful, to substitute $X$ with  a smooth variety, using the resolution of singularity, or (in some cases) even replace $\Bir(X)$ with Aut$(X)$, using the minimal model. However, in the study of $\Bir(X)_d$, such a replacement is not as functional as it appears. For example, the varieties  $X_1$ and $X_2$, in Example \ref{Ex1}, are birationally equivalent. The map $\theta:X_2\dasharrow X_1$ that sends $(x:y:z)$ to $(yz:xz:y^2)$ and $\theta^{-1}:X_1\dasharrow X_2$ that sends $(x:y:z)$ to $(yz:xz:x^2)$ define the birational equivalence. Using these birational maps, we have an isomorphism $\Psi:\Bir(X_1)\to \Bir(X_2)$  given by $\Psi(\varphi)=\theta^{-1}\varphi\theta\in \Bir(X_2) $.  Interestingly, in this example,  $\Psi(\sigma_1)=\sigma_2$. Thus the restriction of $\Psi$ to $\Bir(X_1)_*$ does not define a well-defined map to $\Bir(X_2)_*$.
\end{Remark}
The following proposition has a critical role in the proof of Theorem \ref{Tmain}. It has been already stated in \cite[Proposition 1.2]{HS17}, and its proof is based on the main result of  \cite{DHS12}.
\begin{Proposition}\label{Pinversedegree}   Let $X\subset \mathbb{P}_k^n$ denote a closed reduced and irreducible subvariety, and let $\mathfrak{F}:X\dasharrow  \mathbb{P}_k^m$ be a birational map with image $Y\subset \mathbb{P}_k^m$, admitting a representative of polynomial degree $d\geq 1$.
 Assume that $X$ and $Y$ are non-degenerated.
 Then the polynomial degree of a representative of the inverse map  $\mathfrak{F}^{-1}$ is bounded above by the number
 	$$ B(X,d,n)=2n\bigg[\frac{1}{2}\delta^{2(n+m+1-\Dim(X))^2}+\delta\bigg]^{2^{(\Dim(X)+2)}}$$
 	where $\delta=\max\{d+1,d_0\}$ and  $d_0$ is the maximum degree of a set of minimal generators of the homogeneous defining ideal of $X$ in the given embedding.
\end{Proposition}
In the case where $X=Y$, accordingly $n=m$,  we call the formula in Proposition \ref{Pinversedegree} the inverse bound and denote it by $B(X,d)$.

Next, we recall  some  notation  from commutative algebra.

Let $(A,\mathfrak{m})$ denote a standard graded Noetherian algebra over the field $k$ and let  $I=(f_1,\ldots,f_s)$ stand for a graded ideal of $A$. Set $\mathcal{R}_A(I):=A[It]\subset A[t]$, the Rees algebra of $I$ over $A$.  The special fiber of $I$ is the algebra $\mathcal{F}(I)=\mathcal{R}_A(I)/\mathfrak{m} \mathcal{R}_A(I)$. The Krull dimension of $\mathcal{F}(I)$ is called the {\em analytic spread} of $I$. When $I$ is generated in a single degree, $\mathcal{F}(I)\simeq k[f_1,\ldots,f_s]$.   We say that  $I$ has maximum  analytic spread  if $\ell_A(I)=\max\{\Dim(A),s\}$. For more details on these notations see \cite{SH}.

To begin, we state a computational algebraic lemma.
 \begin{Lemma}\label{Grobner}Let $d$ be an integer and denote by $M_1,\cdots,M_N$ all of the monomials of degree $d$ in the polynomial ring $S=k[x_0,\cdots,x_n]$,  $N=\binomial{n+d}{d}$. Let $\mathfrak{b}$ be a homogenous ideal of $S$. Then  $$V_{\mathfrak{b}}:=\{(a_1,\ldots,a_N)\in \mathbb{P}^{N-1}_k:\sum_{i=1}^N a_iM_i\in \mathfrak{b}\}$$
 is a linear algebraic  subset of $\mathbb{P}^{N-1}_k$. Consequently, $U_{\mathfrak{b}}:=\{(a_1,\ldots,a_N)\in \mathbb{P}^{N-1}_k:\sum_{i=1}^N a_iM_i\not \in \mathfrak{b}\}$ is open in $\mathbb{P}^{N-1}_k$.
 \end{Lemma}
 \begin{proof} We fix a monomial order on $x_0,\ldots,x_n$ and consider a Groebner basis for the ideal $\mathfrak{b}$. If the remainder,  after doing the division algorithm, is zero  then $f\in \mathfrak{b}$. Conversely, if the leading term of the  polynomial $f=\sum_{i=1}^N a_iM_i$  is not divisible by those of the Groebner basis of $\mathfrak{b}$ then $f\not \in \mathfrak{b}$.  Hence, if the leading coefficient of $f$ is not divisible, we set the corresponding coefficient equal to zero and continue with the rest of the polynomial. Notice that this coefficient is a linear equation in terms of $a_i$'s. We then divide $f$ by the polynomials in $\mathfrak{b}$. The quotient is a linear function in terms of $a_i$'s and so that is the remainder of the division.  The result follows by  induction on the number of terms of $f$.
 \end{proof}
The linear structure of the variety $V_{\mathfrak{b}}$ associates  a numerical invariant to any variety $X\subseteq \mathbb{P}^n$.
\begin{Definition}\label{Dindex}  Let $X\subseteq \mathbb{P}^n$ be a projective variety with coordinate ring $R=k[x_0,\cdots,x_n]/\mathfrak{p}$. Let $d$ be an integer and   consider the variety $V_{\mathfrak{p}}$ introduced in Lemma \ref{Grobner}. We call the integer $(\Dim(V_{\mathfrak{p}})+1)$ the $d$'th elimination index of $X$ and denote it by $\epsilon_X(d)$.
\end{Definition}
This notation has a role in the proof of the main theorem. The  special case $\epsilon_X(d)=0$ happens if $X=\mathbb{P}^n$ or if $d$ is less than the minimum degree among the degrees of the generators of the ideal $\mathfrak{p}$.
\medskip

The next openness Lemma is about the locus of those ideals that  are traditionally called {\it ideals of the principal class}.
\begin{Lemma} \label{Elimination}Let $R=k[x_0,\ldots,x_n]/\mathfrak{p}$ be a standard graded ring of dimension $r$. Let $d$ be an integer and denote by $M_1,\cdots,M_N$ all of the monomials of degree $d$ in the polynomial ring $S=k[x_0,\cdots,x_n]$.  For any ${\bf a}=(a_1,\cdots,a_N)\in k^N$, put $f_{{\bf a}}= \sum_{i=1}^N a_iM_i$. Then  for any $1\leq j\leq r$, the set
$$\mathcal{C}_j=\{({\bf a}_1,\ldots,{\bf a}_j)\in \mathbb{P}^{jN-1}: \codim_R(f_{{\bf a}_1},\ldots,f_{{\bf a}_j})=j\}$$
is a non-empty proper open subset of  $\mathbb{P}^{jN-1}$.
\end{Lemma}

 \begin{proof}

 For any integer $t$, $S_t$ is the $k$-vector space generated by monomials of degree $t$.  We denote by $R_t$ the $k$-vector space generated by  the class of the same monomials of $S_t$ modulo $\mathfrak{p}$ (these classes are not necessarily linearly independent).

   We first show that $\mathcal{C}_r$  is open.  Pick $({\bf a}_1,\ldots,{\bf a}_r)\in \mathbb{P}^{rN-1}$ and consider polynomials $f_{{\bf a}_1},\ldots,f_{{\bf a}_r}$. Let's  assume $\mathfrak{p}$ is generated by $p_1,\ldots,p_q$ of respected degrees $d_1,\ldots,d_q$. For any integer $m\geq \max\{d,d_1,\ldots,d_q\}$, consider  the $k$-linear map
    \begin{align*}
T^m:\underbrace{S_{m-d}\oplus\cdots\oplus S_{m-d}}_{r}\oplus S_{m-d_1}\oplus  \cdots \oplus S_{m-{d_q}}&\longrightarrow S_m&&&\\
(g_1,\ldots ,g_r,h_1,\ldots ,h_q)&\mapsto (\sum_{j=1}^rg_jf_{{\bf a}_j}+\sum_{j=1}^qh_jp_j).&
\end{align*}
$T^m$ is surjective if and only if $(x_0,\ldots,x_n)^m\subseteq (f_{{\bf a}_1},\ldots,f_{{\bf a}_r})+\mathfrak{p}$. The latter is equivalent to  $\codim_R(f_{{\bf a}_1},\ldots,f_{{\bf a}_r})=r$.
Passing modulo $\mathfrak{p}$, we have the linear map
 \begin{align*}
\tau^m:\underbrace{R_{m-d}\oplus \cdots\oplus  R_{m-d}}_{r}&\longrightarrow R_m\\
(g_1,\ldots ,g_r)&\mapsto (\sum_{j=1}^rg_jf_{{\bf a}_j}).
\end{align*}
Once more, $\tau^m$ is surjective if and only if   $\codim_R((f_{{\bf a}_1},\ldots,f_{{\bf a}_r}))=r$.

 Using the class of the basis of $S_i$'s to generate $R_i$'s, the presentation matrix of $\tau^m$ is a matrix with ${m+n\choose n}$ rows and ${m-d+n\choose n}r$ columns. One can regard the columns in $r$ partitions. The $i$th partition is described by considering the multiplication of the basis of  $R_{m-d}$ into $f_{{\bf a}_i}$. Hence the entries of the columns of the  $i$'th partition are either $a_{ij}$  for some $j$, or zero.

  Let assume $\dim_k(R_m)=N_m$ and let $I_{N_m}(\tau^m)$ be the ideal generated by the $N_m$-minors of  the representation matrix of $\tau^m$.  Then  $\tau^m$ is surjective if and only if $I_{N_m}(\tau^m)\neq 0$.  Consider the polynomial ring $A=k[a_{ij}:1\leq i\leq r,~~ 1\leq j\leq N]$ and $I_{N_m}(\tau^m)$ as an ideal of $A$.  Besides the standard graded structure of $A$, we consider an $\mathbb{N}^r$-graded structure for $A$ as follows: for all $1\leq j\leq N$ and $1\leq i\leq r$,  $$\deg(a_{ij})=(\underbrace{0,\ldots,\underbrace{1}_{i},0\ldots,0}_r).$$
  That means the columns of the  $i$'th partition are all of a same multi-degree.

  With respect to either grading, $I_{N_m}(\tau^m)$  is a ($\mathbb{N}^r$-)homogenous ideal of $A$.

  Let $$\mathcal{I}=\sum_{m\geq\max\{d,d_1,\ldots,d_r\}}I_{N_m}(\tau^m).$$
Since $A$ is Noetherian, $\mathcal{I}$ is finitely generated. Let  $\mathcal{I}=(Q_1,\ldots,Q_v)$ for some ($\mathbb{N}^r$-)homogenous polynomials $Q_i\in A$. We then have $\mathcal{C}_r=(V(Q_1,\ldots,Q_v))^c$ that is an open Zariski set of  $\mathbb{P}^{rN-1}$.

Now, fix  $j<r$.   Consider the polynomial ring $A'=k[a_{st}:1\leq s\leq j,~~ 1\leq t\leq N]$, let ${\bf a}_i$ to be the sequence  of variables $a_{i,1},\ldots,a_{i,N}$ for all $i$, and define
 $$\mathcal{P}_j:=(Q_i({\bf a}_1,\ldots,{\bf a}_j,\A_{j+1},\ldots,\A_r)\in A': 1\leq i\leq v, \A_t\in \mathbb{P}^{N-1} \text{ and }\{ f_{\A_{j+1}},\ldots,f_{\A_r}\}\in \mathfrak{S})$$
 where $\mathfrak{S}=\{\{f_{\A_{j+1}},\ldots,f_{\A_r}\}: f_{\A_{j+1}},\ldots,f_{\A_r} \text{~~is a part of a system of parameters of~~} R\}$.
We claim  that $\mathcal{C}_j=(V(\mathcal{P}_j))^c\subseteq \mathbb{P}^{jN-1}$.

We first notice that $\mathcal{P}_j$ is a homogenous ideal of $A'$. This is due to the $\mathbb{N}^r$-graded structure of $A$ and homogeneity of  $I_{N_m}(\tau^m)$,   as explained in the previous paragraph. So that $(V(\mathcal{P}_j))^c\subseteq \mathbb{P}^{jN-1}$.

Since $R$ is a quotient of a polynomial ring, $(\A_1,\ldots,\A_j)\in \mathcal{C}_j$ if and only if   $(f_{\A_1},\ldots,f_{\A_j})$ is a part of a system of parameters of $R$ that is $\dim(R/(f_{\A_1},\ldots,f_{\A_j}))=r-j$. Hence there exists $(\A_{j+1},\ldots,\A_r)\in \mathfrak{S}$  such that  $\codim(f_{\A_1},\ldots,f_{\A_j},\ldots,f_{\A_r})=r$; so that there exists $1\leq i\leq v$  with $Q_i({\A_1},\ldots, {\A_j},\ldots, {\A_r})\neq 0$. The latter says that $\mathcal{C}_j\subseteq (V(\mathcal{P}_j))^c$.  The other inclusion is more visible.
\end{proof}

 Now, we pass from "codim" to "grade".
 \begin{Lemma}\label{LG2} Keeping the same notation as in Lemma \ref{Elimination}. The set
 $$G_{2}:=\{({\bf a},{\bf b})\in \mathbb{P}^{2N-1}_k: (f_{{\bf a}},f_{{\bf b}}) \text {~~is a regular sequence in~~} R\} $$ is a Zariski    open subset of $ \mathbb{P}^{2(N-1)}_k$.  $G_2$ is non-empty if and only if $\depth(R)\geq 2$.

  \end{Lemma}
 \begin{proof}
 It is known that for a ring such as $R$ (homomorphic image of Gorenstein ring), the $(S_2)$-locus of $\Spec(R)$ is an open subset, c.f. \cite[12.3.10]{BS}.  Let $V(I)$ be the non-$(S_2)$-locus of $R$,  $I$ is a homogenous ideal of $R$ of codimension at least $2$.

   Let  $\mathcal{A}=\{\mathfrak{q}\in \Spec(R):\codim(\mathfrak{q})=2 \text{ and} \grade(\mathfrak{q})=1\}$. $\mathcal{A}$ may be empty.  If not,   for any $\mathfrak{q}\in \mathcal{A}$, $R_{\mathfrak{q}}$ is not $S_2$, hence $\mathfrak{q}\supset I$, and since $\codim(I)\geq 2$, $\mathfrak{q}$ is a minimal prime over $I$.

   Clearly if $\codim (I)=2$, $\mathcal{A}\subseteq \min (I)$ which is a finite set.

 Recalling the definition of $\mathcal{C}_2$ from Lemma \ref{Elimination},
 $$G_2=\mathcal{C}_2\setminus\{({\bf a},{\bf b})\in \mathbb{P}^{2N-1}_k: \codim(f_{{\bf a}},f_{{\bf b}})=2 \text{~and~} \grade(f_{{\bf a}},f_{{\bf b}})=1\}.$$

 We show that the take-out part is closed.
 \begin{align*}
 \{({\bf a},{\bf b})\in \mathbb{P}^{2N-1}_k: \codim(f_{{\bf a}},f_{{\bf b}})=2 \text{~and~} \grade(f_{{\bf a}},f_{{\bf b}})=1\}&=\\
 \{({\bf a},{\bf b})\in \mathbb{P}^{2N-1}_k: \codim(f_{{\bf a}},f_{{\bf b}})=2 ~\text{and} ~ f_{{\bf a}},f_{{\bf b}}\in \mathfrak{q}~ \text{~for some~} \mathfrak{q}\in \mathcal{A}\}.
 \end{align*}
 By Lemma \ref{Grobner}, $f_{{\bf a}},f_{{\bf b}}\in \mathfrak{q}$ if and only if $({\bf a},{\bf b})\in V_{\mathfrak{p}+\mathfrak{q}}\times V_{\mathfrak{p}+\mathfrak{q}}$ where $\mathfrak{p}$ is the defining ideal of $R$. Therefore
 $$G_2=\mathcal{C}_2\setminus \bigcup_{\mathfrak{q}\in \mathcal{A}}(V_{\mathfrak{p}+\mathfrak{q}}\times V_{\mathfrak{p}+\mathfrak{q}}).$$
This completes the proof,  since $\mathcal{A}$ is a finite set and $\mathcal{C}_2$ is open by Lemma \ref{Elimination}.

  \end{proof}

 At first glance, one may be tempted to prove Lemma \ref{Elimination} and Lemma \ref{LG2} by applying the usual  {\it prime avoidance Lemma} in conjunction with Lemma \ref{Grobner}. The prime avoidance lemma constructs the desired sequences (regular sequence or principal class) inductively, while  in the above Lemmas the desired sequences  are chosen simultaneously. Proving inductively, one needs  more effort to show that the constructed sets are indeed open.

 Next, it is  an openness property related to  the Noether normalization.
 \begin{Lemma}\label{LNn+1}
  Let $k$ be an infinite field and keep the same notation as in Lemma \ref{Elimination}.  Then the following set that   we call  the {\it  Noether Normalization locus}  is  a non-empty open subset of  $\mathbb{P}^{(n+1)N-1}$
  $$\mathcal{N}_{n+1}=\{({\bf a}_0,\ldots,{\bf a}_n)\in \mathbb{P}^{(n+1)N-1}:\ell_R(f_{{\bf a}_0},\cdots,f_{{\bf a}_n})=r\}.$$
   \end{Lemma}

\begin{proof}   We prove by induction on $i\geq r$ that $\mathcal{N}_i$ is a Zariski open where
$$\mathcal{N}_i=\{({\bf a}_0,\ldots,{\bf a}_{i-1})\in \mathbb{P}^{iN-1}:\ell_R(f_{{\bf a}_0},\cdots,f_{{\bf a}_{i-1}})=r\}.$$

First,  we check the case $i=r$. Let $\{\theta_1,\ldots,\theta_r\}\subset R_d$ such that $\Dim(k[\theta_1,\ldots,\theta_r])=r$. Since $k[\theta_1,\ldots,\theta_r]$ is integrally closed and the extension $k[\theta_1,\ldots,\theta_r]\subset R$ is integral, the going-down theorem holds for this extension, in particular, the ideal $(\theta_1,\ldots,\theta_r)R$ has codimension $r$ i.e it is a system of parameters. On the other hand, any system of parameters is analytically independent \cite[Theorem 14.5]{M86}. Therefore
$$\mathcal{N}_r=\{({\bf a}_0,\ldots,{\bf a}_{r-1})\in k^{rN} : \codim_R(f_{{\bf a}_0},\cdots,f_{{\bf a}_{r-1}})=r\}=\mathcal{C}_r.$$
The latter determines  an open subset of $\mathbb{P}^{rN-1}$ by Lemma \ref{Elimination}.

Next, we show that $\mathcal{N}_{r+1}$ is open Zariski. The proof of the  general inductive  pass  is similar.

Let ${\bf a}_1,\ldots,{\bf a}_{r+1}\in k^{N}$ be such that $\ell_R(f_{{\bf a}_1},\ldots,f_{{\bf a}_{r+1}})=r$. Then using the same technique as that of  the proof of the  Noether normalization theorem \cite[Lemma 2, page 262]{M86}, one can show that there exists $(c_1,\ldots,c_{r})\in k^r$ such that $\ell_R(f_{{\bf a}_1}-c_1f_{{\bf a}_{r+1}},\ldots,f_{{\bf a}_r}-c_rf_{{\bf a}_{r+1}})=r$--(here we need an  infinite field).  We repeat this statement in other words: we  denote the points in $k^{(r+1)N},$ $k^{rN}$ and $k^{r(r+1)}$ by matrices of sizes $(r+1)\times N$, $r\times N$ and $r\times (r+1)$, moreover,  $M=(m_1,\ldots,m_N)^t$ the $N\times 1$ matrix of the monomials of degree $d$ in $R$. Then for any $(r+1)\times N$-matrix $A=({\bf a}_{ij})$ with $\ell_R(AM)=r$, there exists an $r\times {(r+1)}$-matrix $C$ such that $CA\in \mathcal{N}_r$.

Conversely, if for an $(r+1)\times N$-matrix $A=({\bf a}_{ij})$  there exists an $r\times {(r+1)}$-matrix $C$ such that $CA\in \mathcal{N}_r$. Then $r=\ell(CAM)\leq \ell(AM)\leq \ell(M)=r$. So that the rows of $A$ constitute a point of $k^{(r+1)N}$ that belongs to $\mathcal{N}_{r+1}$.
In brief
\begin{equation}\label{nr+1}
\mathcal{N}_{r+1}=\{A\in  k^{(r+1)N}: \text{~~there exists~~}  C\in k^{r(r+1)} \text{~~with~~} CA\in \mathcal{N}_r\}.
\end{equation}
Restating the above equality, any $r\times(r+1)$ matrix $C$ determines a linear map $\varPhi_C:k^{r+1}\to k^r$.  $N$ copies such a map define a linear map
 $$\bigoplus^N\varPhi_C:k^{(r+1)N}\to k^{rN}.$$
The equivality (\ref{nr+1}) states that
$$\mathcal{N}_{r+1}=\bigcup_C (\bigoplus^N\varPhi_C)^{-1}(\mathcal{N}_r).$$
The linearity of $\bigoplus^N\varPhi_C$ guarantees that $\mathcal{N}_{r+1}$ is an open Zariski subset of $\mathbb{P}^{(r+1)N-1}$.
The inductive pass of the induction argument  follows by showing that
$$\mathcal{N}_{r+j}=\bigcup_C (\bigoplus^N\varPhi_C)^{-1}(\mathcal{N}_{r+j-1}).$$

Yet, in the following paragraph, we state a constructive proof of the openness of $\mathcal{N}_{r+1}$. Consider three sets of independent variables
$$ \begin{pmatrix}
  Z_{1,1}&\ldots&\ldots&Z_{1,N}\\
  \cdots&\cdots&\cdots&\cdots\\
  Z_{r,1}&\ldots&\ldots&Z_{r,N}
 \end{pmatrix},
 \quad
 \begin{pmatrix}
  X_{1,1}&\ldots&\ldots&X_{1,N}\\
  \cdots&\cdots&\cdots&\cdots\\
  X_{r,1}&\ldots&\ldots&X_{r,N}\\
  X_{r+1,1}&\ldots&\ldots&X_{r+1,N}
 \end{pmatrix},
 \quad
 \begin{pmatrix}
  Y_{1,1}&\ldots&Y_{1,r+1}\\
  \cdots&\cdots&\cdots\\
  Y_{r,1}&\ldots&Y_{r,r+1}
 \end{pmatrix},
 $$
  and  set $\ZZ_i=(Z_{i,1},\ldots, Z_{i,N})$, $\XX^i=(X_{i,1},\ldots,X_{r+1,i})^t$ and $\YY_i=(Y_{i,1},\ldots,\YY_{i,r+1})$.
   Assume that $\mathbb{P}^{rN-1}\setminus \mathcal{N}_r=V(\mathfrak{n}_r)$ where $\mathfrak{n}_r\subset k[\ZZ_1,\ldots,\ZZ_r]$ is a homogenous ideal.
   For any $F\in \mathfrak{n}_r$, set
   $$g_F(\YY_1,\ldots,\YY_r,\XX^1,\ldots,\XX^N)=F((\YY_1\XX^1,\ldots,\YY_1\XX^N),\ldots,(\YY_r\XX^1,\ldots,\YY_r\XX^N))).$$
 Let $\mathfrak{n}=(g_F: F \in  \mathfrak{n}_r)$ be the homogenous ideal in  $k[\YY_1,\ldots,\YY_r,\XX^1,\ldots,\XX^N]$.
 The projection $\pi:\Proj(k[\YY_1,\ldots,\YY_r,\XX^1,\ldots,\XX^N])\to \Proj(k[\XX^1,\ldots,\XX^N])$ is a closed map. Hence, there is a homogenous ideal $\mathfrak{n}_{r+1}\subset k[\XX^1,\ldots,\XX^N]$ with $V(\mathfrak{n}_{r+1})=\pi(V(\mathfrak{n}))$.

 The explanation of $\mathcal{N}_{r+1}$ in (\ref{nr+1}) shows that  $$\mathbb{P}^{(r+1)N-1}\setminus \mathcal{N}_{r+1}=V(\mathfrak{n}_{r+1}),$$  that completes the proof.
 \end{proof}

 We are now ready to state and prove  the main theorem.  We adapt some of the notation of   \cite[Lemma 2.4]{BF13}.  As above,  we  fix an integer $d$ and denote by $M_1,\ldots,M_N$  all of the monomials of degree $d$ in the polynomial ring $S=k[x_0,\cdots,x_n]$.
 Consider the projective space $\mathbb{P}_k^{(n+1)N-1}$. Any point ${\bf a}$ of $\mathbb{P}_k^{(n+1)N-1}$ is an  $(n+1)$-uple  $({\bf a}_0,\ldots,{\bf a}_n)$ with ${\bf a}_i\in \mathbb{A}^N_k$. For each index ${\bf a}_i$ we construct a polynomial
 $$f_{{\bf a}_i}= \sum_{j=1}^N ({\bf a}_i)_jM_j.$$
 We denote $W_d$ to be the set of equivalence classes of nonzero $(n+1)$-uple $(f_0,\ldots,f_n)$ of homogeneous polynomials $f_i \in S$ of degree $d$ where $(f_0,\ldots,f_n)$ is equivalent to $(\lambda f_0,\ldots,\lambda f_n)$ for any $\lambda\in k\setminus{\{0\}}$. The equivalence class of $(f_0,\ldots,f_n)$ will be denoted by $(f_0:\ldots :f_n)$. It is clear that  $\mathbb{P}^{(n+1)N-1}$   is isomorphic to $W_d$ via the correspondence:
 \begin{align*}
 \theta:\mathbb{P}^{(n+1)N-1}&\to W_d\\
 {\bf a}=({\bf a}_0,\ldots,{\bf a}_n)&\mapsto f_{{\bf a}}=(f_{{\bf a}_0}:\cdots:f_{{\bf a}_n}).
 \end{align*}

We define $H_d\subseteq W_d$ to be the set of elements $f=(f_0,\ldots,f_n)\in W_d$ such that the rational map
\begin{align*}
\psi_f:X&\dasharrow X\\
(x_0:\cdots:x_n)&\mapsto  (f_0(x_0,...,x_n) : \cdots: f_n(x_0,...,x_n))
\end{align*}
  is a well-defined birational  map of {\it polynomial degree }$d$.

  $\tilde{H}_d\subseteq H_d$ is the set of elements  $f=(f_0,\ldots,f_n)\in H_d$ such that $\psi_f$ is birational of {\it clear polynomial degree} $d$--($\tilde{H}_d$ corresponds to $H_{d,d}$ in \cite[Lemma 2.13 and Proposition 2.15]{BF13}).

\begin{Theorem}\label{Tmain}  Let $d$ be an integer,  $k$  an infinite field and $X\subseteq \mathbb{P}^n$ an irreducible projective non-degenerated variety.   Keeping the   the same notation as above,
\begin{itemize}
\item[(i)]{$H_d$ and $\tilde{H}_d$ are  quasi-projective varieties; and}
\item[(ii)]{$\Bir(X)_d$ has the structure of  a constructible algebraic set. }
\end{itemize}
\end{Theorem}
\begin{proof} (i) The first paragraph of the proof  follows the argument of \cite[Lemma 2.4]{BF13}. Let $R=k[x_0,\cdots,x_n]/\mathfrak{p}$ be the coordinate ring of $X$. Let  $\mathfrak{p}=(p_1,\ldots,p_t)$ and $r=\Dim(R)=\Dim(X)+1$. Let $d'=B(X,d)$ be the inverse bound introduced in Proposition \ref{Pinversedegree}. We denote by $Y'\subseteq W_{d'}\times W_d$ the set consisting of elements $(g, f )$ such that $h := (g_0(f_0,\ldots,f_n),\ldots, g_n(f_0,\ldots,f_n))$  and $h' := (f_0(g_0,\ldots,g_n),\ldots, f_n(g_0,\ldots,g_n))$ are  multiples of the identity modulo $\mathfrak{p}$; i.e., $h_ix_j-h_jx_i\in \mathfrak{p}$  and $h'_ix_j-h'_jx_i\in \mathfrak{p}$ for all $i,j$.

In oder to $f:X\dasharrow X$ and $g:X\dasharrow X$ define well-defined maps, the coordinates must satisfy $p_i(f_0,\ldots,f_n)\in \mathfrak{p}$ and $p_i(g_0,\ldots,g_n)\in \mathfrak{p}$ for all $i$.

Let $$\mathcal{F}_i:=\{({\bf a}_0,\ldots,{\bf a}_n)\in W_d:p_i(f_{{\bf a}_0},\ldots,f_{{\bf a}_n})\in \mathfrak{p}\}$$ and $$\mathcal{G}_i:=\{({\bf a}'_0,\ldots,{\bf a}'_{n})\in W_{d'}:p_i(f_{{\bf a}'_0},\ldots,f_{{\bf a}'_n})\in \mathfrak{p}\}.$$

With the same proof as that of  Lemma \ref{Grobner} one can show that   $\mathcal{F}_i \subseteq W_d$ and $\mathcal{G}_i\subseteq W_{d'}$ are closed varieties (not necessarily linear).  Similarly, $Y'$ is a closed subset of $W_{d'}\times W_d$. Hence
$$Y:=Y'\cap (\bigcap_{i=1}^t \mathcal{F}_i \times W_{d'})\cap (W_d\times \bigcap_{i=1}^t \mathcal{G}_i )$$
is a closed subset of $W_{d'}\times W_d$. Now, applying the principal theorem of the elimination theory,  the image of $Y$ by the projection $\pi_2:W_{d'}\times W_d\to W_d$ is closed in $W_d$.

Next, we look for $f=(f_0,\ldots,f_n)\in W_d$ such that the  rational map $f:X\dasharrow X$ is dominant. Being dominant  means that the Krull dimension of the image of $f$ is $r-1$. The coordinate ring of the closure of the image of $f$ is $\Proj(\mathcal{F}_R(I))$ (see definitions  after Proposition \ref{Pinversedegree}) wherein $I$ is the ideal generated by the classes of $f_0,\ldots,f_n$ in $R$. Therefore the map $f$ is dominant if and only if $\ell_R(I)=r$.  According to Lemma \ref{LNn+1}, this defines the set $\mathcal{N}_{n+1}$. Therefore,
$$H_d=\{h\in W_d: \psi_h:X \dasharrow X \text{~~is birational of polynomial degree~~} d\}=\pi_2(Y)\cap \theta(\mathcal{N}_{n+1}).$$
That is an open subset of the algebraic set  $\pi_2(Y)$ thence a quasi-projective variety.

Next,  we show that $\tilde{H}_d$ is a quasi-projective variety. $\tilde{H}_d$ is the intersection of  $H_d$ with those birational maps whose base ideal is of  grade at least $2$.  The regular sequence of length $2$ may not be among the generators, however it is inside a linear combination of them. Hence we need to work with
$$U_2=\{({\bf a}_0,\ldots,{\bf a}_n)\in \mathbb{P}^{(n+1)N-1 }:\sigma({\bf a}_0,\ldots,{\bf a}_n)\in G_2\times \mathbb{A}^{(n-1)N} \text{~for some~} \sigma\in {\rm GL}_{n+1}(k)\}.$$
That is
$$U_2=\bigcup_{\sigma\in {\rm GL}_{n+1}(k) }\sigma^{-1}(G_2\times \mathbb{A}^{(n-1)N}).$$
Since  $G_2$ is an open  subset,  by Lemma \ref{LG2}, $U_2\subseteq \mathbb{P}^{(n+1)N-1}$ is open, as well.

\medskip
We now have $\tilde{H}_d=\pi_2(Y)\cap \theta(\mathcal{N}_{n+1}\cap U_2)$ thence $\tilde{H}_d$ is  a  quasi-projective variety.
\medskip

(ii)  In the case where $X=\mathbb{P}^n$, the proof of this part is to show that there is a bijection between $\tilde{H}_d$ and $ \Bir(X)_d$.  However, when $X\neq \mathbb{P}^n$, even if  a map $f=(f_0,\ldots,f_n):X\dasharrow X$ belongs to $\tilde{H}_d$, the defining polynomials of $f$ in $S$ are not uniquely defined. In other words, two sets of polynomials  in $S$ say $f_0,\ldots,f_n$ and $f'_0,\ldots,f'_n $,  define the same map over $X$ if $f_i-f'_i\in \mathfrak{p}$ for all $i$.

To remedy this situation, we return to Lemma \ref{Grobner} and Definition \ref{Dindex}.
Let $$s=N-\epsilon_X(d).$$
 Then the ideal of definition of the linear variety  $V_{\mathfrak{p}}:=\{(a_1,\ldots,a_N)\in \mathbb{P}^{N-1}_k:\sum_{i=1}^N a_iM_i\in \mathfrak{p}\}$  is determined by  $s$ linear equations $l_1,\ldots,l_s$.
 Let $\mathcal{L}:k^N\to k^s$ be the linear map defined by $l_1,\ldots,l_s$. By repeating $\mathcal{L}$ on the diagonal, we extend it to a map $k^{(n+1)N}\to k^{(n+1)s}$ thence to a map $\mathcal{D}: W_d\dasharrow \mathbb{P}^{(n+1)s-1}$.

 More precisely, let $y_1,\ldots,y_N$ be the coordinates of $\mathbb{A}_k^N$, $l_i=\sum_{j=1}^N l_{ij}y_j$ for $i=1,\ldots,s$  and  $(f_{{\bf a}_0}:\ldots :f_{{\bf a}_n})\in W_d$. Then
  $$\mathcal{D}(f_{{\bf a}_0}:\cdots :f_{{\bf a}_n})=(l_1({\bf a}_0):\cdots:l_s({\bf a}_0):\cdots:l_1({\bf a}_n):\cdots:l_s({\bf a}_n)).$$
  Let $\mathcal{D}'=\mathcal{D}|_{\tilde{H}_d }$.
\
Recalling the definition of the clear polynomial degree of a birational map, we have the natural  map
\begin{align*}
\psi:{\tilde H}_d&\longrightarrow \Bir(X)_d\\
h&\mapsto  \psi_h.
\end{align*}
  The map $\psi$ induces a well-defined map $\Phi$, according to the following triangle
\begin{equation}\label{diagram}
\xymatrix{
	& \tilde{H}_d \ar^{\psi}[rd]                         &                   \\
	\Image(\mathcal{D}')\ar@^{<--} [ru]^{\mathcal{D}'} \ar[rr]^{\Phi} &   & \Bir(X)_d.  \\
}
\end{equation}

We verify that $\Phi$ is bijective.

Injectivity: Suppose that $f=(f_{{\bf a}_0}:\cdots:f_{{\bf a}_n})$ and $f'=(f_{{\bf a}'_0}, \ldots, f_{{\bf a}'_n})$ are two elements of $\tilde{H}_d$ such that $\Phi (\mathcal{D}'(f))=\Phi(\mathcal{D}'(f'))$. Hence $\psi_f=\psi_{f'}\in \Bir(X)_d$.

Since the base ideal of $\psi_f$ and $\psi_{f'}$, as ideals of $R$, have grade at least $2$ over $R$, they differ by multiplication into a constant. Thus, if  the difference depends on a constant multiplication,  lifting to the (quasi-)projective space  $\tilde{H}_d$, we have $(f_{{\bf a}_0}:\ldots:f_{{\bf a}_n})=(f_{{\bf a}'_0}:\ldots:f_{{\bf a}'_n})$.  The other point is that lifting from $\Bir(X)_d$ to  ${\tilde H}_d$ depends on  the natural ring map $S\to R$. Hence, the two representatives may differ by the equations of $\mathfrak{p}$, that is, (looking at the representatives as elements of $S$)  $f_{{\bf a}_i}-f_{{\bf a}_i'}\in \mathfrak{p}$ for all $i$.   Hence ${\bf a}_i-{\bf a}_i'\in V_{\mathfrak{p}}$. Therefore, $l_j({\bf a}_i-{\bf a}_i')=0$ for all $i$ and $j$. It then follows that $$\mathcal{D}(f_{{\bf a}_0}-f_{{\bf a}'_0}, \ldots, f_{{\bf a}_n}-f_{{\bf a}'_n})=0.$$
Since $\mathcal{D}$ is a linear map, we get $$\mathcal{D}'(f)=\mathcal{D}'(f').$$

Surjectivity: Let $h=(f_{{\bf a}_0}:\cdots:f_{{\bf a}_n}):X\dasharrow X$ be a birational map of clear degree degree $d$. Since $\grade_R(f_{{\bf a}_0},\ldots,f_{{\bf a}_n})\geq 2$, we have $(f_{{\bf a}_0}:\cdots:f_{{\bf a}_n}) \in \theta(U_2)$. Since $h$ is dominant $\Dim(\Proj(k[f_{{\bf a}_0},\ldots,f_{{\bf a}_n}]))=\Dim(X)$ hence $\ell_R(f_{{\bf a}_0},\ldots,f_{{\bf a}_n})=r$; so that $(f_{{\bf a}_0}:\cdots:f_{{\bf a}_n})\in \theta( \mathcal{N}_{n+1})$, and finally since the birational map $h$ has an inverse of polynomial degree at most $d'=B(X,d)$ by Proposition \ref{Pinversedegree}, $(f_{{\bf a}_0}:\cdots:f_{{\bf a}_n}) \in \pi_2(Y)$; accordingly $(f_{{\bf a}_0}:\cdots:f_{{\bf a}_n})\in \pi_2(Y)\cap \theta(\mathcal{N}_{n+1}\cap U_2)=\tilde{H}_d$.  Therefore  $\Phi(\mathcal{D}'((f_{{\bf a}_0}:\cdots:f_{{\bf a}_n})))=h$.

The bijectivity between $\Image(\mathcal{D}')$ and  $\Bir(X)_d$ ensues that the latter inherits the same topological structure as any one of the $\Image(\mathcal{D}')$. Since $\tilde{H}_d$ is quasi-projective,  by part (i), then  Chevally's Theorem implies that $\Image(\mathcal{D}')$ is constructible,  and so is $\Bir(X)_d$.
\end{proof}
\begin{Corollary}\label{C1} With the same token  as Theorem \ref{Tmain}, if $\epsilon_X(d)=0$, then $\Bir(X)_d$ has the structure of a quasi-projective variety.  In particular, if  $d$ is less than the minimum degree of the generators of $\mathfrak{p}$, then $\Bir(X)_d$ has the structure of a quasi-projective variety.
\end{Corollary}
\begin{proof} $\epsilon_X(d)=0$  implies that $s=N$, in the course of the proof \ref{Tmain}. Hence the map $\mathcal{D}'$ is the identity map. So that $\psi$ is a bijection.
\end{proof}

It is worth mentioning one point about the proof of the main theorem.

\begin{Remark}  One may  use the Jacobian criterion to detect dominant maps. However, this criterion is characteristic sensitive; so the proof thereof is only valid for fields of characteristic zero.  For instance
 \begin{align*}
\sigma :\mathbb{P}^2&\dasharrow \mathbb{P}^2\\
(x:y:z)&\mapsto (x^2:xy:yz)
\end{align*}

is birational over any field. However,   over fields of characteristic $2$, the Jacobian matrix has rank $2$ instead of $3$.

We are not aware that  there exists a  Jacobian-like  criterion to detect the dominancy of a map over $X\neq \mathbb{P}^n$, even in the case of zero characteristic.

\end{Remark}

redpajama_set_name: RedPajamaArXiv


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Adonis aka (Dennis) Padua is a graphic design artist with over 10 yrs experience in illustration, logo branding and design. He's worked with several up and coming artists and brands that you may be already be familiar with. Whenever he's not sketching away on his 22" HD Cintiq, he enjoys watching cartoons, sipping iced lattes and rearranging his extensive Transformers collection.
Disclaimer: Due to the increased cost of living and rough economy, Dennis is unable to accept payments in gum.


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Speedsolving.com Wiki
From Speedsolving.com Wiki
Welcome to the SpeedSolving.com Wiki!
Learn about speedsolving the Rubik's Cube and other twisty puzzles on this wiki. Built to compliment our community of 35,000+ solvers, this wiki has tutorials, methods, records, and articles on puzzle solving. The 1,459 articles contain information on getting faster as well as on various aspects of speedcubing and the Speedsolving.com community.
Speedsolving.com Wiki is made by people like you. Sign up to join our 3,627 registered users and contribute to our 52,712 edits.
OLL
CFOP
Introduction to Speedcubing
4x4x4 Parity Algorithms
Today's featured article
Feliks Zemdegs
Today's featured article is Feliks Zemdegs.
Feliks Zemdegs is an Australian speedcuber from Melbourne, Australia. He is widely considered to be the most successful competitive cuber in history, being the first ever to hold over one hundred world records since the start of his cubing career. Only recently have other people begun to catch up, with Max Park being the first to (briefly) break Felik's 7-year streak of holding the world record average for the Rubik's cube in 2017.
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Speedsolving.com Wiki needs your help ! We want to create a various community who talks about cube-related things. For more information about the purpose of this Wiki and how to edit, please visit the about page.
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Today's featured picture
Dave3457 made this image to explain how does a commutator affects cubies.
A 3x3x3 centre piece
Centers don't move related to each other.
Cubing terminology
Puzzle notations
History of cubing
The 3x3x3 cube (also known as "Rubik's cube" and "magic cube") was invented by Ernő Rubik in 1974, and was quickly unveiled in the early 1980s.
This is a mechanical and geometric puzzle composed of with 26 elements (called cubies), which can be moved with a system of axes, that is hidden in the center of the cube. Each of the six faces is covered by 9 stickers, among six solid colours (traditionally being white, yellow, orange, red, blue, and green). The system of axes enables each layer to turn independently, thus mixing up the colours. For the puzzle to be solved, each face must be a solid colour.
It is said to be the world's best-selling toy, with over 350 millions cubes sold worldwide as of 2009.
It has inspired many widely available variations such the 2×2×2, the 3×3×3, the 4×4×4, the 5×5×5, larger sizes ones, the Pyraminx (tetrahedron), the Skewb Diamond (octahedron), the Megaminx (dodecahedron) or the Dogic (icosahedron). All of these items belong to a broad category of puzzles commonly referred to as "twisty puzzles". Some twisty puzzles are shape-shifting (such as the Square One) or custom-built (such as shape modifications of existing mechanisms).
Many subjects have been built around cubing:
Cubing competitions
Blindfolded solving
Feet solving
Fewest Moves
One-Hand solving
Optimal solution
Puzzle softwares
Puzzle solving
Solving Variations
Blindfolded solving (BLD)
One-Handed solving (OH)
Fewest Moves (FMC)
Solving with feet
Cubing methods and substeps
Common substeps :
Cross, F2L, LL, OLL, PLL, CMLL, PBL, CLL, ELL, CxLL, ZBLL, more...
Common cube states :
Scrambled, Solved, F2L, Eoline, 2x2x2 block, 2x2x3 block, more...
2x2x2 beginner methods, as the Ortega method
2x2x2 speedsolving methods, as the EG method
3x3x3 beginner methods, as The Beginner method
3x3x3 speedsolving methods, as the CFOP method
Blindsolving methods, Fewest Moves methods, Experimental methods, Big Cubes methods
Fastest Videos for each Method
Puzzle Hardware
2x2x2 cubes
Bigcubes
Puzzle Types:
2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 7x7x7, Pyraminx, Megaminx, Square-1, Magic, more...
Competitive Cubing
Introduction to Speedcubing for Beginners
Road to Sub-15 Second Averages
Competitions General
List of Official World Records
List of Unofficial World Records
List of Puzzle Shops
List of Cubing Methods
Utilities and Cubers' Hangouts
Web-based software
Cubing code on GitHub
Speedcubing.com
Speedsolving Forums
#rubik
Category:Contents: the top-level category of this wiki
Algorithm Database
Notable cubers
Cubing websites
YouTube cubers
About Speedsolving.com Wiki
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SpeedSolving Forum


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Review: MahlerFest XXXII, Kelly Dean Hansen, 5/23/19
"In addition to playing the demanding piano part in the Elgar, (Korevaar) took on a challenge few pianists would dare: one of Franz Liszt's legendary arrangements of the Beethoven symphonies for solo piano. These arrangements are notoriously difficult, among the most challenging things for any pianist to tackle. In Korevaar's hands, the Fourth Symphony lost none of its identity despite the absence of the orchestra. This is a testament to both Liszt's skill as a transcriber and Korevaar's pianism. The perpetual motion finale—certainly not idiomatic for the piano—was breathtaking. In fact, this may rank as Korevaar's finest Boulder performance in several years, and that is saying something." | full review
Review: PERRACHIO 9 Poems – 25 Preludes • David Korevaar, Gramophone, May 2019
"Kudos to David Korevaar for bringing Perrachio's inventive and pianistically resourceful piano music to light and playing it so wonderfully well. Excellent sound…" | full review
Review: PERRACHIO Piano Music • David Korevaar, MSR MS1710, Gramophone, March 2019
"The Italian composer and pianist Luigi Perrachio was born in Turin in 1883 and first learned music as a child from his father, an amateur pianist. After gaining a law degree in Turin he studied music in Bologna before travelling to Paris, where he first heard the music of Ravel and Debussy, who greatly influenced him. Perrachio then settled in Turin, where he taught at the Liceo Musicale from 1925 to 1955. As a writer, pianist and conductor, he was a propagandist for contemporary music and an educational reformer. Of his modest output, in the Debussy-Ravel tradition, the 25 Preludes (1927) for piano are best known. He was important in promoting new through his activities as a performer and organizer, especially with a group he put together, the 'Double Quintet of Turin'. He was far from prolific as a composer, only as a consequence of significant arm-twisting from his friends. Innate shyness, combined with an aristocratic sense of composing, led him to publish only a small part of his works (especially those for piano and lyrics) and starting from the mid-twenties to confine himself to a sort of voluntary isolation. On this beautifully produced CD, the brilliant American musician David Korevaar gives intelligent, thoughtful and dynamic performances of piano pieces by Luigi Perrachio – bringing this deightful assured music to a much wider audience. 'A musical epiphany.'"
Review: PERRACHIO Nove Poemetti, 25 Preludi • David Korevaar, MSR 1710, Fanfare (Jerry Dubins), May/June 2019
"This is altogether a fascinating album and the music a real treat for the ears. It would be reason enough to applaud David Korevaar for discovering these works and enabling us to hear them on record for the first time, but a standing ovation is in order, along with the applause for playing of such beauty and artistry of such penetrating insight. Urgently recommended." | full review
Review: PERRACHIO Nove Poemetti, 25 Preludi • David Korevaar, MSR 1710, Fanfare (Huntley Dent), May/June 2019
"(Korevaar's) touch is worthy of Debussy and Ravel, his phrasing captures the melodic line with an unerring ear, and he incorporates the quicksilver Impressionist gestures with delicacy. The result is that the poetry intended by Perrachio comes through perfectly." | full review
Review: PERRACHIO Nove Poemetti, 25 Preludi • David Korevaar, MSR 1710, Fanfare (Colin Clarke), May/June 2019
"Quite a find here—literally! Korevaar's disc…is a major addition to the catalog; in fact, it is nothing short of revelatory. Luigi Perracchio could hardly ask for a better interpreter. One fervently hopes this opens the door to more recordings of Perrachio's music." | full review
Pianist David Korevaar Discusses the Music of Composer Luigi Perrachio, Fanfare, May/June 2019
"This recording features the world premiere of two Perrachio compositions, the Nove Poemetti (1917–20), and 25 Preludi (1927). The Nove Poemetti (Nine Poems) are a collection of short works composed between 1917 and 1920. Most are accompanied in the score by a brief verse that complements the music. The 25 Preludes are in the tradition of like works by Bach, Chopin, Scriabin, and other composers…David Korevaar proves himself a superb advocate. His playing is technically brilliant, with a wonderful plasticity of phrasing and colors, and at all times displaying a captivating flow and momentum.
…As in the case of the Poems, I found the Preludes to be highly attractive and compelling listening. That is a credit both to the composer, and of course, Korevaar's superb playing and interpretation. It's clear that discovering, learning, and performing Luigi Perrachio's music has been a labor of love for David Korevaar." | full article
Review: LOWELL LIEBERMANN: Piano Works, Vol 3 – David Korevaar – MSR, Fanfare, Nov/Dec 2018
"Pianist David Korevaar, to whom Liebermann has dedicated at least one work, displays his considerable gifts in these readings. He dashes off the technical demands of some of these works with aplomb, while caressing the simple and quiet sections to bring out every last bit of their poetic nature. Here is a pianist with palpably clear musical gifts and intelligence, and one who should be widely heard." | full review
Review: Lowell Liebermann Piano Music, Vol 3, Gramophone, October 2018
"Needless to say, one needs boundless resources of technique, tone, color, and musical imagination to do Liebermann's keyboard aesthetic justice. David Korevaar possesses these qualities in spades and manages to create an individual character with each selection. Both the present disc and Korevaar's two earlier Liebermann volumes brilliantly reflect this pianist's probing individuality and firm commitment to the material." | full review